11
syllabus
syllabus
rref
efer
erence
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Topic:
• Applied statistical
analysis
In this
In this
cha
chapter
pter
11A Informal description of
chance
11B
Single event probability
11C Relative frequency
11D Modelling probability
11E
Long-run proportion
Introduction
A major outdoor rock concert is to be staged in your local area. The date that has been suggested for the con-cert is the last weekend in February. Your job is to report back to the organisers on the suitability of this date.
You have been told that the organisers will lose a lot of money if the concert is rained out. You have also been told that if the temperature rises to above 35°C, there will be a strain on medical services as many in the crowd will collapse from heat exhaus-tion.
You must comment on the likelihood of either of these events occurring. If either is likely, you will need to suggest a more suitable date for the concert.
Informal description of chance
You have booked a ski holiday to Thredbo for the middle of July. What is the chance that there will be enough snow on the ground for you to ski? There is no exact answer to this question, but by looking at the amount of snow in Thredbo during July over past years, we know that there is a very good chance that there will be enough snow to ski again this year.
You can then say that it is very likely that you will be able to ski during July at Thredbo. Terms such as ‘very likely’, ‘almost certain’, ‘unlikely’ and ‘fifty-fifty’ are used in everyday language to describe the chance of an event occurring. For the purposes of probability, an event is the outcome of an experiment a statistician is interested in. They can describe an outcome as a possible result to the probability experiment.
Imagine you are playing a board game and it is your turn to roll the die. To win the game you need to roll a number less than 7. If you roll one die you must get a number less than 7. The chance of this event occurring is described as certain.
When an event is certain to occur, the probability of that event occurring is 1.
Now consider an impossible situation.
In a board game you have one last throw of the die. To win you must roll a 7. We know that this cannot be done. We would say that this is impossible.
When an event is impossible, the probability of the event is 0.
The chance of any event occurring will often be somewhere between being certain and impossible and we use a variety of terms to describe where the chance lies in this range as shown in the figure at right.
We use these terms based on our general knowledge of the world, the total possible outcomes and how often an event occurs.
impossible very unlikely unlikely fifty-fifty probable certain almost certain 1
0
You will need to use these terms to describe events that are more likely to occur than others.
The term frequency refers to how often an event occurs. We use our knowledge about possible outcomes to order outcomes from the most frequent to the least frequent. Describe the chance of each of the following events occurring.
a Tossing a coin and it landing Heads. b Rolling a 6 with one die.
c Winning the lottery.
d Selecting a spot card (ace, 2, 3 . . ., 10) from a standard deck.
THINK WRITE
a There is an equal chance of the coin landing Heads and Tails.
a The chance of tossing a head is fifty-fifty.
b There is only one chance in six of rolling a 6.
b It is unlikely that you will roll a 6.
c There is only a very small chance of winning the lottery.
c It is very unlikely that you will win the lottery.
d There are more spot cards than picture cards in a deck.
d It is probable that you will select a spot card.
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Mrs Graham is expecting her baby to be born between July 20 and 26. Is it more likely that her baby will be born on a weekday or a weekend?
THINK WRITE
There are 5 chances that the baby will be born on a weekday and 2 chances that it will be born on a weekend.
It is more likely that Mrs Graham’s baby will be born on a weekday.
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A card is chosen from a standard deck. List the following outcomes in order from least likely to most likely.
selecting a picture card selecting an ace selecting a diamond selecting a black card
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In the above examples, we have been able to calculate which event is more likely by counting the number of ways an event may occur. This is not always possible. In some cases we need to use general knowledge to describe the chance of an event occurring.
Consider the following probability problems.
Problem A
‘The letters of the alphabet are written on cards and one card is selected at random. Which letter has the greater chance of being chosen, E or Q?’
Each letter has an equal chance of being chosen because there is one chance that E will be chosen and one chance that Q will be chosen.
Problem B
‘Stacey sticks a pin into a page of a book and she writes down the letter nearest to the pin. Which letter has the greater chance of being chosen, E or Q?’
Problem B is more difficult to answer because each letter does not occur with equal frequency. However, we know from our experience with the English language that Q will occur much less often than most other letters. We can therefore say that E will occur more often than Q.
This is an example of using knowledge of the world to make predictions about which event is more likely to occur. In this way, we make predictions about everyday things such as the weather and which football team will win on the weekend.
THINK WRITE
There are 12 picture cards in the deck. There are 4 aces in the deck.
There are 13 diamonds in the deck.
There are 26 black cards in the deck. The order of events in ascending order of likeli-hood:
selecting an ace selecting a picture card selecting a diamond selecting a black card.
1 2 3 4
During the 2001 NRL season, the Brisbane Broncos played the New Zealand Warriors in Round 6. The Brisbane Broncos had won the last 9 matches between the two teams. Which team would be more likely to win?
THINK WRITE
Brisbane Broncos have a better record than the New Zealand Warriors.
Brisbane Broncos would be more likely to win, based on their previous results. (Footynote: The Warriors won the game 13–12. Brisbane was more likely to win the game but nothing in foot-ball is certain.)
4
This is one example of past results being used to predict future happenings. There are many other such examples.
Informal description of
chance
1 Describe the chance of each of the following events occurring, using an appropriate probability term.
a Selecting a ball with a double-digit number from a bag with balls numbered 1 to 40
b Selecting a female student from a class with 23 boys and 7 girls
c Selecting a green marble from a barrel with 40 blue marbles and 30 red marbles
d Choosing an odd number from the numbers 1 to 100
2 For each of the events below, describe the chance of it occurring as impossible, unlikely, even chance (fifty-fifty), probable or certain.
a Rolling a die and getting a negative number
b Rolling a die and getting a positive number
c Rolling a die and getting an even number
d Selecting a card from a standard deck and getting a red card
e Selecting a card from a standard deck and getting a spot (numbered) card
f Selecting a card from a standard deck and getting an ace
g Reaching into a moneybox and selecting a 30c piece
h Selecting a blue marble from a bag containing 3 red, 3 green and 6 blue marbles Weather records show that it has rained on Christmas Day 12 times in the last 80 years. Describe the chance of it raining on Christmas Day this year.
THINK WRITE
It has rained only 12 times on the last 80 Christmas Days. This is much less than half of all Christmas Days.
It is unlikely that it will rain on Christmas Day this year.
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remember
1. The chance of an event occurring ranges from being certain to impossible. 2. (a) An event that is certain has a probability of 1.
(b) An event that is impossible has a probability of 0.
3. There are many terms that we use to describe the chance of an event occurring, such as improbable, unlikely, fifty-fifty, likely and probable.
4. Sometimes we can describe the chance of an event occurring by counting the possible outcomes, while other times we need to rely on our general knowledge to make such a description.
remember
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3 Give an example of an event which has a probability that could be described as:
4 Is it more likely that a person’s birthday will occur during a school term or during the school holidays?
5 For each event on the left, state whether it is more likely, less likely or equally likely to occur than the event on the right.
a Fine weather Christmas Day Wet weather Christmas Day
b A coin landing Heads A coin landing Tails
c Rolling a total of 3 with two dice Rolling a total of 7 with two dice
d Winning a raffle made up of 50 tickets Winning a raffle made up of 200 tickets
e Winning a prize in the Lotto draw Not winning a prize in the Lotto draw
6 A die is thrown and the number rolled is noted. List the following events in order from least likely to most likely.
Rolling an even number Rolling a number less than 3 Rolling a 6
Rolling a number greater than 2
7 Write the following events in order from least to most likely. Winning a raffle with 5 tickets out of 30
Rolling a die and getting a number less than 3
Drawing a green marble from a bag containing 4 red, 5 green and 7 blue marbles Selecting a court card (ace, king, queen, jack) from a standard deck
Tossing a coin and having it land Heads
8 Before meeting in the cricket World Cup in 1999, Australia had beaten Zimbabwe in 10 of the last 11 matches. Who would be more likely to win on this occasion?
9 Which of the following two runners would be expected to win the final of the 100 metres at the Olympic Games?
Carl Bailey — best time 9.92 s and won his semi-final Ben Christie — best time 10.06 s and 3rd in his semi-final Give an explanation for your answer.
10
A stack of 26 cards has the letters of the alphabet written on them. Vesna draws a card from that stack. The probability of selecting a card that has a vowel written on it could best be described as:
11
Which of the following events is the most likely to occur?
A Selecting the first number drawn from a barrel containing 20 numbered marbles B Selecting a diamond from a standard deck of cards
C Winning the lottery with one ticket out of 150 000
D Drawing the inside lane in the Olympic 100-metre final with eight runners E Correctly selecting the first drawn ball in Gold Lotto
a certain b probable c even chance
d unlikely e impossible.
A impossible B unlikely C even chance D probable E almost certain
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The ski season opens on the first weekend of June. At a particular ski resort there has been sufficient snow for skiing on that weekend on 32 of the last 40 years. Which of the following statements is true?
A Snow on the opening day of the ski season is impossible. B It is unlikely to snow at the opening of the ski season this year.
C There is a fifty-fifty chance that it will snow at the opening of the ski season this year.
D It is probable that it will snow at the opening of the ski season this year. E It is certain to snow at the opening of the ski season this year.
13 On a production line, light globes are tested to see how long they will last. After testing 1000 light globes it is found that 960 will burn for more than 1500 hours. Wendy purchases a light globe. Describe the chance of the light globe burning for more than 1500 hours.
14 Of 12 000 new cars sold last year, 1500 had a major mechanical problem during the first year. Edwin purchased a new car. Describe the chance of Edwin having a major mechanical problem in the first year.
15 During an election campaign, 2000 people were asked for their voting preferences. One thousand said that they would vote for the government. If one person is chosen at random, describe the chance that they would vote for the government.
Single event probability
So far in this chapter we have discussed the chances of certain events occurring. In doing so, we used informal terms such as probable and unlikely. However, while these terms give us an idea of whether something is likely to occur or not, they do not tell us how likely it is. To do this, an accurate way of stating the probability is needed.
We stated earlier that the chance of any event occurring was somewhere between impossible and certain. We also said that:
• if an event is impossible the probability was 0 • if an event is certain the probability was 1.
It therefore follows that the probability of any event must lie between 0 and 1 inclusive.
A probability of an event is a number which describes the chance of that event
occur-ring. All probabilities are calculated as fractions but can be written as fractions, decimals or percentages. Probability is calculated using the formula:
What will the weather be?
As you work through this chapter your task is to analyse the suitability of the last weekend in February to stage a major outdoor rock concert. Based only on your own knowledge of the weather pattern in your local area, describe in words: • the chance of the concert being rained out
• the chance that the temperature will rise to 35°C on this particular weekend. m
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P event( ) number of favourable outcomes total number of outcomes
The total number of favourable outcomes is the number of different ways the event can occur, while the total number of outcomes is the number of elements in the sample space. The sample space, S, is a list of all the possible outcomes enclosed in curled brackets.
Consider the case of tossing a coin. If we are calculating the probability that it will land Heads, there is 1 favourable outcome out of a total of 2 possible outcomes. Hence we can then write P(Heads) = . This method is used to calculate the probability of any single event.
In the above example the fraction could be simplified to .
Some questions have more than one favourable outcome. In these cases, we need to add together each of these outcomes to calculate the number of which are favourable. Zoran is rolling a die. To win a game, he must roll a number greater than 2. List the sample space and state the number of favourable outcomes.
THINK WRITE
There are 6 possible outcomes. S = {1, 2, 3, 4, 5, 6}
The favourable outcomes are to roll a 3, 4, 5 or 6.
There are 4 favourable outcomes.
1 2
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---Andrea selects a card from a standard deck. Find the probability that she selects an ace.
THINK WRITE
There are 52 cards in the deck (total number of outcomes).
There are 4 aces (number of favourable outcomes).
Write the probability. P(ace) =
1
2
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---In a barrel there are 6 red marbles, 2 green marbles and 4 yellow marbles. One marble is drawn at random from the barrel. Calculate the probability that the marble drawn is red.
THINK WRITE
There are 12 marbles in the barrel (total number of outcomes).
There are 6 red marbles in the barrel (number of favourable outcomes).
Write the probability. P(red) =
1
2
3 12---6
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Some questions do not require us to calculate the entire sample space, only the sample space for a small part of the experiment. We can often do this in number questions such as worked example 10. We may need to concern ourselves with only one digit rather than the whole number, meaning we do not need to write out the whole sample space. On a bookshelf there are 4 history books, 7 novels, 2 dictionaries and 5 sporting books. If I select one at random, what is the probability that the one chosen is not a novel?
THINK WRITE
There are 20 books on the shelf (total number of outcomes).
Seven of these books are novels, meaning that 13 of them are not novels (number of favourable outcomes).
Write the probability. P(not a novel) =
1
2
3 1320
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The digits 1, 3, 4, 5 are written on cards and these cards are then used to form a four-digit number. Calculate the probability that the number formed is:
a even
b greater than 3000.
THINK WRITE
a If the number is even the last digit must be even.
a
There are four cards that could go in the final place (total number of outcomes).
Only one of these cards (the 4) is even (number of favourable outcomes).
Write the probability. P(even) =
b If the number is greater than 3000, then the first digit must be a 3 or greater.
b
There are four cards that could go in the first place.
Three of these cards are a 3 or greater.
Write the probability. P(greater than 3000) =
1
2
3
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2
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4 34
---10
Single event probability
1 A coin is tossed at the start of a cricket match. Manuel calls Heads. List the sample space and the number of favourable outcomes.
2 For each of the following probability experiments, state the number of favourable outcomes.
a Rolling a die and needing a 6
b Rolling two dice and needing a total greater than 9
c Choosing a letter of the alphabet and it being a vowel
d The chance a baby will be born on the weekend
e The chance that a person’s birthday will fall in summer
3 For each of the following probability experiments, state the number of favourable outcomes and the total number of outcomes.
a Choosing a red card from a standard deck
b Selecting the winner of a 15 horse race
c Selecting the first ball drawn in a Lotto draw (The balls are numbered 1 to 44.)
d Winning a raffle with 5 tickets out of 1500
e Selecting a yellow ball from a bag containing 3 yellow, 4 red and 4 blue balls
4 A coin is tossed. Find the probability that the coin will show Tails.
remember
1. The sample space is the list of all possible outcomes in a probability experiment.
2. The event space is a list of all favourable outcomes to a probability experiment. 3. The probability of an event is calculated using the formula:
P(event) = number of favourable outcomes total number of outcomes
---remember
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5 A regular die is cast. Calculate the probability that the uppermost face is:
a 6
b 1
c an even number
d a prime number
e less than 5
f at least 5.
6 A barrel contains marbles with the numbers 1 to 45 written on them. One marble is drawn at random from the bag. Find the probability that the marble drawn is:
7 Many probability questions are asked about decks of cards. You should know the cards making up a standard deck.
A card is chosen from a standard deck. Find the probability that the card chosen is:
8 A bag contains 12 counters: 7 are orange, 4 are red and 1 is yellow. One counter is selected at random from the bag. Find the probability that the counter chosen is:
9 The digits 2, 3, 5 and 9 are written on cards. One card is then chosen at random. Find the probability that the card chosen is:
10 In a bag of fruit there are 4 apples, 6 oranges and 2 pears. Larry chooses a piece of fruit from the bag at random but he does not like pears. Find the probability that Larry does not select a pear.
11 The digits 2, 3, 5 and 9 are written on cards. They are then used to form a four-digit number. Find the probability that the number formed is:
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A die is rolled. The probability that the number on the uppermost face is less than 4 is:
a 23 b 7 c an even number
d an odd number e a multiple of 5 f a multiple of 3
g a number less than 20 h a number greater than 35 i a square number.
a the ace of diamonds b a king c a club
d red e a picture card f a court card.
a yellow b red c orange.
a the number 2 b the number 5 c even
d odd e divisible by 3 f a prime number.
a even b odd c divisible by 5
d less than 3000 e greater than 5000.
A B C D E
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When a die is rolled, which of the following outcomes does not have a probability equal to ?
A The number on the uppermost face is greater than 3. B The number on the uppermost face is even.
C The number on the uppermost face is odd. D The number on the uppermost face is at least a 3. E The number on the uppermost face is a prime number.
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A card is chosen from a standard deck. The probability that the card chosen is a court card is:
15
When a card is chosen from a standard deck, which of the following events is most likely to occur?
16 One thousand tickets are sold in a raffle. Craig buys five tickets.
a One ticket is drawn at random. The holder of that ticket wins first prize. Find the probability of Craig winning first prize.
b After the first prize has been drawn, a second prize is drawn. If Craig won first prize, what is the probability that he now also wins second prize?
17 A lottery has 160 000 tickets. Janice buys one ticket. There are 3384 cash prizes in the lottery.
a What is the probability of Janice winning a cash prize?
b If there are 6768 consolation prizes of a free ticket for being one number off a cash prize, what is the probability that Janice wins a consolation prize?
c What is the probability that Janice wins either a cash prize or a consolation prize?
18 A number is formed using all five of the digits 1, 3, 5, 7 and 8. What is the probability that the number formed:
19 Write down an example of an event which has a probability of:
20 A three-digit number is formed using the digits 2, 4 and 7.
a Explain why it is more likely that an even number will be formed than an odd number.
b Which is more likely to be formed: a number less than 400 or a number greater than 400?
A B C D E
A choosing a seven B choosing a club
C choosing a picture card D choosing a black card E choosing a spot card
a begins with the digit 3? b is even? c is odd?
d is divisible by 5? e is greater than 30 000? f is less than 20 000?
a b c .
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---Comparing theoretical probabilities
with experimental results
In this activity, we compare the probability of certain events to practical results. You may be able to do a simulation of these activities on a spreadsheet.
1 Tossing a coin
a If we toss a coin P(Heads) = . Therefore, if you toss a coin, how many Heads would you expect in:
i 4 tosses? ii 10 tosses? iii 50 tosses? iv 100 tosses? Now toss a coin 100 times and record the number of Heads after:
i 4 tosses? ii 10 tosses? iii 50 tosses? iv 100 tosses. Combine your results with the rest of the class. How close to 50% is the total number of Heads thrown by the class?
2 Rolling a die
When you roll a die, what is the probability of rolling a 1? The probability for each number on the die is the same.
Roll a die 120 times and record each result in the table below.
How close are the results to the results that were expected?
3 Rolling two dice
Roll two dice and record the total on the faces of the two dice. Repeat this 100 times and complete the table below.
Do you notice anything different about the results of this activity, compared to the others?
1 2
---Number Occurrences Percentage of throws
1 2 3 4 5 6
Number Occurrences Percentage of throws
Relative frequency
You are planning to go skiing on the first weekend in July. The trip is costing you a lot of money and you don’t want your money wasted on a weekend without snow. So what is the chance of it snowing on that weekend? We can use past records only to estimate that chance.
If we know that it has snowed on the first weekend of July for 54 of the last 60 years, we could say that the chance of snow this year is very high. To measure that chance, we calculate the relative frequency of snow on that weekend. We do this by dividing the number of times it has snowed by the number of years we have examined. In this case, we can say the relative frequency of snow on the first weekend in July is 54 ÷ 60 = 0.9. The relative frequency is usually expressed as a decimal or percentage and is calculated using the formula:
Relative frequency =
In this formula, a trial is the number of times the probability experiment has been conducted.
The formula for relative frequency is similar to that for probability.
The term relative frequency refers to actual data obtained, but the term probability
generally refers to theoretical data unless experimental probability is specifically stated.
Experimental or theoretical?
We have just been calculating the probability of certain events occurring. Now consider the following statements made about the weather conditions for the proposed rock concert.
• The weather on the day of the concert will either be wet or dry. Therefore, the probability of rain on the day of the concert is . There is a 50% chance that the organisers will lose their money.
• The coldest possible temperature in your area is 10°C and the hottest possible temperature is 40°C. Therefore the probability that the temperature will rise to 35°C is .
Are these statements correct? Why or why not? 1 2
---5 30
---number of times an event has occurred number of trials
---The weather has been fine on Christmas Day in Brisbane for 32 of the past 40 Christmas Days. Calculate the relative frequency of fine weather on Christmas Day.
THINK WRITE
Write the formula. Relative frequency =
Substitute the number of fine Christmas Days (32) and the number of trials (40).
Relative frequency =
Calculate the relative frequency as a decimal. = 0.8
1
number of times an event has occurred number of trials
---2 3240
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The relative frequency is used to assess the quality of products. This is done by finding the relative frequency of defective products.
Relative frequencies can be used to solve many practical problems.
A tyre company tests its tyres and finds that 144 out of a batch of 150 tyres will withstand 20 000 km of normal wear. Find the relative frequency of tyres that will last 20 000 km. Give the answer as a percentage.
THINK WRITE
Write the formula. Relative frequency =
Substitute 144 (the number of times the event occurred) and 150 (number of trials).
Relative frequency =
Calculate the relative frequency. = 0.96
Convert the relative frequency to a percentage. = 96%
1
number of times an event has occurred number of trials
---2 144150
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A batch of 200 light globes was tested. The batch is considered unsatisfactory if more than 15% of globes burn for less than 1000 hours. The results of the test are in the table below.
Determine if the batch is unsatisfactory.
Continued over page No. of hours No. of globes
less than 500 4
500–750 12
750–1000 15
1000–1250 102
1250–1500 32
more than 1500 35
THINK WRITE
Count the number of light globes that burn for less than 1000 hours.
31 light globes burn for less than 1000 hours.
Write the formula. Relative frequency =
1
2
number of times an event has occurred number of trials
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Relative frequency
1 At the opening of the ski season, there has been sufficient snow for skiing for 37 out of the past 50 years. Calculate the relative frequency of sufficient snow at the begin-ning of the ski season.
2 A biased coin has been tossed 100 times with the result of 79 Heads. Calculate the relative frequency of the coin landing Heads.
3 Of eight Maths tests done by a class during a year, Peter has topped the class three times. Calculate the relative frequency of Peter topping the class.
4 Farmer Jones has planted a wheat crop. For the wheat crop to be successful farmer Jones needs 500 mm of rain to fall over the spring months. Past weather records show that this has occurred on 27 of the past 60 years. Find the relative frequency of:
a sufficient rainfall
b insufficient rainfall.
5 Of 300 cars coming off an assembly line, 12 are found to have defective brakes. Calculate the relative frequency of a car having defective brakes. Give the answer as a percentage.
THINK WRITE
Substitute 31 (number of times the event occurs) and 200 (number of trials).
Relative frequency =
Calculate the relative frequency. = 0.155
Convert the relative frequency to a percentage.
= 15.5%
Make a conclusion about the quality of the batch of light globes.
More than 15% of the light globes burn for less than 1000 hours and so the batch is unsatisfactory.
3 200---31
4 5
6
remember
1. The relative frequency is used to estimate the probability of an event.
2. The relative frequency, usually expressed as a decimal or percentage, is a figure that represents how often an event has occurred.
3. The relative frequency is calculated using the formula:
Relative frequency = number of times an event has occurred. number of trials
---remember
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6 A survey of 25 000 new car buyers found that 750 had a major mechanical problem in the first year of operation. Calculate the relative frequency of:
a having mechanical problems in the first year
b not having mechanical problems in the first year.
7 On a production line, light globes are tested to see how long they will last. After testing 1000 light globes, it is found that 960 will burn for more than 1500 hours. Wendy purchases a light globe. What is the relative frequency that the light globe will:
a burn for more than 1500 hours?
b burn for less than 1500 hours?
8
A study of cricket players found that of 150 players, 36 batted left handed. What is the relative frequency of left-handed batsmen?
9
Four surveys were conducted and the following results were obtained. Which result has the highest relative frequency?
A Of 1500 P-plate drivers, 75 had been involved in an accident. B Of 1200 patients examined by a doctor, 48 had to be hospitalised.
C Of 20 000 people at a football match, 950 were attending their first match. D Of 50 trucks inspected, 2 were found to be unroadworthy.
E Of 300 drivers breath tested, 170 were found to be over the legal limit.
10 During an election campaign 2000 people were asked for their voting preferences. One thousand and fifty said that they would vote for the government, 875 said they would vote for the opposition and the remainder were undecided. What is the relative frequency of:
a government voters?
b opposition voters?
c undecided voters?
11 Research over the past 25 years shows that each November there is an average of two wet days on Sunnybank Island. Travelaround Tours offer one-day tours to Sunnybank Island at a cost of $150 each, with a money back guarantee against rain.
a What is the relative frequency of wet November days as a percentage?
b If Travelaround Tours take 1200 bookings for tours in November, how many refunds would they expect to give?
12 An average of 200 robberies takes place each year in the town of Amiak. There are 10 000 homes in this town.
a What is the relative frequency of robberies in Amiak?
b Each robbery results in an average insurance claim of $20 000. What would be the minimum premium per home that the insurance company would need to charge to cover these claims?
A 0.24 B 0.36 C 0.54 D 0.64 E 0.76
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13 A car maker recorded the first time that its cars came in for mechanical repairs. The results are in the table below.
The assembly line will need to be upgraded if the relative frequency of cars needing mechanical repair in the first year is greater than 25%. Determine if this will be neces-sary.
14 For the table in question 13 determine, as a percentage, the relative frequency of:
a a car needing mechanical repair in the first 3 months
b a car needing mechanical repair in the first 2 years
c a car not needing mechanical repair in the first 3 years.
15 A manufacturer of shock absorbers measures the distance that its shock absorbers can travel before they must be replaced. The results are in the table below.
The relative frequency of the shock absorber lasting is 0.97 for a certain guaranteed distance. What is the maximum distance the manufacturer will guarantee so that the relative frequency of the shock absorbers lasting is greater than 0.97?
16 A soccer team plays 40 matches over a season and the results (wins, losses and draws) are shown below.
W W W D L L L D W L W D L D
W W L L L D W W D L L W W
W L D L D D L W W W D D L
a Put this information into a table showing the number of wins, losses and draws.
b Calculate the relative frequency of each result over a season.
Time taken No. of cars
0–3 months 5
3–6 months 12
6–12 months 37
1–2 years 49
2–3 years 62
more than 3 years 35
No. of kilometres No. of shock absorbers
0–20 000 1
20 000–40 000 2
40 000–60 000 46
60 000–80 000 61
80 000–100 000 90
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Modelling probability
We have now seen examples where probability can be calculated using our knowledge of possible outcomes and can be estimated using past results. We are able to compare the theoretical probability and actual results using a simulation.
While it may be possible to count the number of outcomes in the event space and the number of successful outcomes, often it is preferable to calculate probabilities
experi-mentally, and to compare this result with the theoretical result. For example, we could
toss a coin 100 times and, perhaps, find that 56 of the tosses resulted in a Head. There-fore the experimental probability = , although the theoretical probability = . It can be shown that the more times the experiment is repeated the closer the experimental
Researching relative frequencies
Choose one of the topics below (or another of your choice) and calculate the relative frequency of the event. Most of the information needed can be found from books or the Internet.
1 Examine weather records and find out the relative frequency of rain on New Year’s Eve in Brisbane.
2 Choose your favourite sporting team. Find the relative frequency of them winning over the past three seasons.
3 Find the relative frequency of the stock market rising for three consecutive days.
4 Check the NRL or AFL competitions and find the relative frequencies of win, loss and draw for each team.
Applying relative frequency
You are now in a position to look at the actual likelihood of either of the given conditions applying on the proposed date of the rock concert.
1 Use the Internet or other sources to find the weather conditions in your area on the last weekend of February for the past 20 years.
2 What is the relative frequency of rain on this weekend?
3 What is the relative frequency of the temperature reaching 35°C on this weekend?
4 Based on these results would you recommend that the concert proceed on the weekend suggested? Explain why or why not.
5 Can you find the weekend during the year where these two conditions are least likely to occur?
56 100
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---result should be to the theoretical one. Therefore, if a coin is tossed 1000 times (for example), the result should be closer to . As with most things in probability, there are no guarantees, but the larger the number of repetitions of an experiment (or of a trial), the more likely the experimental probability is close to the theoretical one.
It is often difficult to complete an actual probability experiment. For example, it may take too long to roll a die 1000 times and record the results. To make this quicker and easier we may perform a simulation.
A simulation is when random numbers are used to obtain results that will follow the
same pattern as a real life experiment. For example, a computer or graphics calculator could randomly select the number 1 or 2 to simulate the toss of a coin, where 1 rep-resents a Head and 2 reprep-resents a Tail.
Consider the following result when rolling a die 12 times: 5, 3, 1, 4, 4, 3, 6, 1, 6, 4, 1, 3.
This sequence of numbers can be treated as a set of random numbers, whose poss-ible values are 1, 2, 3, 4, 5, 6.
A deck of playing cards could be numbered from 1 to 52; by drawing a card, then replacing it in the deck, shuffling the deck and drawing another card, a sequence of random numbers between 1 and 52 would be generated. There are several other ways of generating random numbers; can you think of any?
There are three general rules that a set of random numbers must follow.
Rule 1. The set must be within a defined range (for example, whole numbers between 1 and 10, or decimals between 0 and 1). The range need not have a definite starting and finishing number, but in most cases it does. For example, when simulating rolling a die we must set a lower limit of 1 and an upper limit of 6.
Rule 2. All numbers within the defined range must be possible outcomes (for example using a die but not counting 4s is not a proper sequence of random numbers between 1 and 6).
Rule 3. No random number in a sequence can depend on any of the previous (or future) numbers in the sequence. (This is difficult to prove but is assumed with dice, spinners and so on.)
These rules can make it extremely difficult to obtain a proper set of random numbers in practice. Furthermore, to generate many random numbers, say 1000, by rolling dice could take a long time! Fortunately, computers can be used to generate random numbers for us. (Technically, they are known as pseudo-random numbers, because rules 2 and 3 above cannot be rigorously met.)
A useful function on the TI–83 is randInt (lowest value, highest value, number of values).
To use this function,
1. Press , select PRB and 5:randInt(.
2. Type the required arguments. For example, to produce four values between 1 and 6, type 1,6,4).
3. Press .
A scrollable list is produced on the home screen (use arrow keys to move within it). In the case of a large number of values, it is convenient to store these to a list. To store a list of 100 values between 1 and 6 to L1,
1 2
---Graphics Calculator
Graphics Calculator
tip!
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Random number generation
CASIO
Random numbers
MATH
1. Enter randInt(1,6,100).
2. Press and [L1].
3. Press .
4. To view the list in table form, press , select 1:Edit, and use the up and down
keys to move within the table.
STO → 2nd ENTER
STAT
Use a sequence of 20 random numbers to simulate rolling a die 20 times. Record your results in a frequency table.
THINK WRITE/DISPLAY
On a T1-83 graphics calculator, go to the
randInt function ( , PRB, 5:randInt(.
Enter randInt(1,6,20), press and
[L1]. Press .
Press , select 1:Edit and count
the number of each of the outcomes 1 to 6 in L1.
Note: Naturally, the values shown in the
screen diagrams may differ each time this sequence is repeated.
Enter this information in a table as shown.
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MATH
2 STO →
2nd ENTER
3 STAT
4 Die values 1 2 3 4 5 6 Frequency 2 4 6 2 5 1
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Generate 50 random numbers between 5 and 10, and sketch a histogram of the results.
THINK DISPLAY
Press , select PRB and
5:randInt.
Enter randInt(5,10,50), press
and [L1]. Press .
Press [STAT PLOT], and set up
Plot 1 as shown.
Arrange suitable WINDOW values and
press . Continued over page
1 MATH
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3 2nd 4
GRAPH
15
Modelling probability
1 Use a sequence of 6 random numbers to simulate rolling a die 9 times. Record your results in a frequency table.
2 Generate 100 numbers between 10 and 20. Produce a histogram of your results.
3 Produce a histogram showing the results of 1000 simulated rolls of a die.
4 a Suggest how a graphics calculator could be used to simulate the tossing of a coin.
b Generate 10 ‘coin tosses’ using this method.
c Sketch a frequency histogram for your results.
5 Repeat question 4 for 100 coin tosses. What do you notice about the histogram?
6 A mini-lottery game may be simulated as follows. Each game consists of choosing two numbers from the whole numbers 1 to 6. The cost to play one game is $1. A particular player always chooses the numbers 1 and 2. A prize of $10 is paid for both numbers correct. No other prizes are awarded.
a Simulate a game by generating 6 random numbers between 1 and 6. Do this 20 times (that is, ‘play’ 20 games of lotto). How many times does the player win?
b What is the player’s profit/loss based on the simulation?
7 Simulate 20 tosses of two dice (die 1 and die 2). How many times did die 1 produce a lower number than die 2? (Hint: Generate two lists of 20 values between 1 and 6.)
8 A board game contains two spinners pictured below.
a Simulate 10 spins of each spinner.
b How many times does the total (that is, both spinners’ numbers added) equal 5?
c How often is there an even number on both spinners?
d How often does the highest possible total occur?
THINK DISPLAY
Press to count the number of each value.
Note: Naturally, the illustrated values may differ from the ones you generate.
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Long-run proportion
Unless we happen to ‘know’ the true probability of an event, a simulation as outlined previously provides us with a method of finding the ‘experimental’ probability. In fact, how do we know that the probability of getting a Tail when tossing a fair coin is really
? The answer to this lies in the belief that if we tossed a coin enough times, the pro-portion of Tails would be close to, if not exactly, . However, there are no guarantees; different simulation experiments will yield different results, but it is the sum total of all
coin-toss simulations that would be likely to yield a result close to 0.5.
With software or calculator simulations it is possible to ‘toss’ a coin hundreds, if not thousands, of times to see if this occurs. By recording the ratio of Tails to ‘coins tossed’ we can compute the experimental probability or long-run proportion. Note that the term proportion is similar, in this chapter, to probability.
The Maths Quest CD-ROM contains files that simulate coin tossing and die rolling.
The screen on the following page shows the frequency of each number rolled after simulating 10, 100 or 1000 throws of a die.
Random choice
1 Use your graphics calculator to generate 50 random numbers between 1 and 100.
2 Display the results in the table below.
3 Display the results in a frequency histogram.
4 Now ask 50 people chosen at random to select a number between 1 and 100. Place the results in a similar table and display the results in a histogram.
5 Which set of results appears to be more random in nature? Explain your answer with reference to the shape of the histograms you have drawn.
Random number Frequency
1–10
11–20
21–30
31–40
41–50
51–60
61–70
71–80
81–90
91–100
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---EXCEL Spreadshe et Coin tossing
Finding the long-run proportion
Similarly, the screen below shows the number of heads and tails obtained after simu-lating the toss of a coin 100 times. The number of tosses can be changed and a new simulation can be produced by pressing F9.
By starting with a sequence of Heads/Tails, we can determine the long-run proportion for coin tosses. This is best shown by an example.
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Given the following sequence of 20 coin tosses, determine its long-run proportion for Heads.
H T H H T T T H T H T T H H T H H T H H
THINK WRITE
Create 2 columns.
Column 1 counts coin tosses. Column 2 counts Heads.
That is, after 1 toss, there was 1 Head. After 2 tosses there was still 1 Head. After 3 tosses there were 2 Heads. After 4 tosses there were 3 Heads . . . and so on.
Divide column 2 by column 1. The result is shown in column 3, rounded to 3 decimal places. (a) The long-run proportion for 20
tosses is given by the last number in column 3, namely 0.550.
(b) Observe how the proportion varies wildly after the first few tosses, but begins to ‘stabilise’ as the number increases.
(c) Note that 20 tosses are not enough to give a long-run proportion very close to the ‘true’ probability of 0.5. 1
Column 1 (coin toss)
Column 2 (heads)
Column 3 (proportion)
1 1 1 ÷ 1 = 1.000 2 1 1 ÷ 2 = 0.500 3 2 2 ÷ 3 = 0.667 4 3 3 ÷ 4 = 0.750 5 3 3 ÷ 5 = 0.600 6 3 3 ÷ 6 = 0.500 7 3 3 ÷ 7 = 0.429 8 4 4 ÷ 8 = 0.500 9 4 4 ÷ 9 = 0.444 10 5 5 ÷ 10 = 0.500 11 5 5 ÷ 11 = 0.455 12 5 5 ÷ 12 = 0.417 13 6 6 ÷ 13 = 0.462 14 7 7 ÷ 14 = 0.500 15 7 7 ÷ 15 = 0.467 16 8 8 ÷ 16 = 0.500 17 9 9 ÷ 17 = 0.529 18 9 9 ÷ 18 = 0.500 19 10 10 ÷ 19 = 0.526
20 11 11 ÷ 20 =0.550
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Another application of long-run proportion is to use experimental data to estimate the true probability of an event. Consider an application in the game of cricket.
This implies that the batsman has a probability of making a run-scoring stroke of about 0.57. It is significant to note that this proportion didn’t change much after the first 6 or 7 innings. Although this might not be the best way to measure a batsman’s ‘effectiveness’ in cricket, it is similar to the method used in baseball to measure ‘batting average’, a method which has been used for over 100 years.
It is not necessary to calculate long-run proportion at all stages of the experiment. Often it is sufficient merely to take the data at the end of the experiment and perform the appropriate division.
A possible measure of a batsman’s effectiveness in test cricket is the number of times he makes a run-scoring stroke as a proportion of balls faced. Consider this set of data from 10 innings over 5 test matches for well-known cricketer Nalla Redrob. Calculate the final long-run proportion of scoring strokes to balls faced.
Innings 1 2 3 4 5 6 7 8 9 10
Run-scoring strokes
34 1 67 38 12 15 47 69 43 18
Balls faced 62 6 107 87 29 19 75 119 67 31
THINK WRITE
Create 4 columns.
(a) Put the innings number in column 1.
(b) Put the cumulative (total) number of run-scoring strokes in column 2.
(c) Put the cumulative (total) number of balls faced in column 3. Complete column 4. (a) Divide the number in
column 2 by the number in column 3. (b) Enter this result in
column 4. This is the long-run proportion for run-scoring strokes to balls faced.
(c) Round the answer to 3 decimal places.
1
Column 1 (innings)
Column 2 (run-scoring
strokes)
Column 3 (balls faced)
Column 4 (proportion)
1 34 62 34 ÷ 62 = 0.548 2 34 + 1 = 35 62 + 6 = 68 35 ÷ 68 = 0.515 3 35 + 67 = 102 68 + 107 = 175 102 ÷ 175 = 0.583 4 102 + 38 = 140 175 + 87 = 262 140 ÷ 262 = 0.534 5 140 + 12 = 152 262 + 29 = 291 152 ÷ 291 = 0.522 6 152 + 15 = 167 291 + 19 = 310 167 ÷ 310 = 0.539 7 167 + 47 = 214 310 + 75 = 385 214 ÷ 385 = 0.556 8 214 + 69 = 283 385 + 119 = 504 283 ÷ 504 = 0.562 9 283 + 43 = 326 504 + 67 = 571 326 ÷ 571 = 0.571 10 326 + 18 = 344 571 + 31 = 602 344 ÷ 602 =0.571 2
17
It is only after a very large number of trials, say 100 000, that we could be confident that the experimental probability was close to the theoretical probability.
The screen view of a spreadsheet showing 1000 simulated coin tosses is shown below. A die which is suspected of being biased (unfair) is tossed 800 times and it is observed that a 6 appeared 205 times. Calculate the long-run proportion and comment on the result.
THINK WRITE
Calculate the long-run proportion. In this case divide the number of 6s by the total.
Long-run proportion =
Long-run proportion= 0.256
Compare with the ‘theoretical’ result. We expect about of the results to be 6s, or about 133 out of 800. This proportion is 0.167. The experimental result indicates that the die is likely to be unfair with 6s appearing too often.
1 205
800
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remember
1. In the long run, the experimental probability obtained using random numbers becomes closer to the true probability. This is known as long-run proportion. 2. The greater the number of trials, the closer the experimental probability
approaches the theoretical probability.
Long-run proportion
1 Given the following sequence of 20 rolls of a die, determine its long-run proportion for sixes.
2, 5, 6, 3, 3, 2, 4, 4, 5, 5, 3, 6, 4, 3, 3, 5, 3, 3, 2, 1
2 A possible measure of a batsman’s effectiveness in test cricket is the number of times he makes a run-scoring stroke as a proportion of balls faced. Consider the data from 12 innings over 6 test matches for cricketer Ian Toppig. Calculate the final long-run proportion of scoring shots to balls faced.
3 a A travelling salesperson records her daily success rate at selling vacuum cleaners. On Day 1 she visited 28 houses and sold 7 vacuum cleaners. Calculate the long-run proportion of sales to houses visited for Day 1.
b Now calculate the long-run proportion of sales to houses visited for each day and comment on whether she is improving her ability as a salesperson.
4
A vacuum cleaner salesman on his first 5 days sold the following number of cleaners: 5, 2, 4, 5, 3. He restricted himself to 20 visits per day. His long-run proportion of sales is:
5
Two statisticians perform a simulation of tossing a single coin. Statistician A does a simulation of 100 tosses, while Statistician B simulates 1000 tosses. Which statistician will get a result closest to 0.5 for the proportion of Heads?
A Definitely Statistician A B Definitely Statistician B C They will get the same result. D More likely to be Statistician B E None of the above
6 A coin suspected of being biased is tossed 600 times and it is observed that a Head appeared 425 times. Calculate the long-run proportion and comment on the result.
Innings 1 2 3 4 5 6 7 8 9 10 11 12
Scoring shots 23 26 45 9 23 34 56 37 18 23 31 46
Balls faced 57 57 89 19 46 72 100 68 31 50 66 89
Day 1 2 3 4 5 6 7 8 9 10
Sales 7 4 6 8 16 12 7 12 12 13
Houses visited 28 19 21 25 42 29 22 31 27 26
A B C D E
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7 A student answers 200 questions from a mathematics textbook and answers 156 of them correctly. Estimate the probability that the student will answer the next question correctly.
8 Another student answers only 145 questions from the same textbook as in question 7 and answers 119 of them correctly. Is this student more effective at answering math-ematics questions?
9 A die is tossed to determine whether it is a fair one and the results are recorded. Calculate the proportion of each value and comment on the results.
10 Three different bread shops recorded the number of loaves of bread sold and the number ordered for 1 month. Which of the three was the most effective? Support your statement with mathematical evidence.
11 Use a spreadsheet or other means to calculate the long-run proportion for:
a 1000 coin tosses
b 1000 die rolls.
12 A cricketer wishes to measure her effectiveness against left- and right-handed bowlers and records the following data over 2 years.
a Is she more likely to be caught by left- or right-handers?
b Is she more likely to be caught or bowled?
c Is she more likely to be not out against left- or right-handers?
d Use the results from a to c to comment on the cricketer’s measure of effectiveness against left- and right-handed bowlers.
13 A method of estimating the number of trout in a lake works as follows. Take a number of trout from another lake, tag them and put them in the lake. A week or so later catch the same number of trout from this lake. Calculate the proportion of tagged fish caught to tagged fish thrown in. The method uses this same proportion as the number of untagged fish to untagged fish caught in the lake. Suppose 100 fish are tagged and thrown in. A week later 100 fish are caught and 7 of them are tagged. Estimate the number of fish in the lake (excluding the tagged ones).
Value 1 2 3 4 5 6
Frequency 457 590 189 623 234 444
Bread shop 1 2 3
Loaves sold 67 167 267
Loaves ordered 89 199 309
Innings Innings caught Innings bowled Innings not out
Left-handers 6 4 4
Right-handers 24 26 6
EXCEL Spreadshe et Coin tossing
EXCEL Spreadshe et Die rolling
Work
Informal description of chance
• The chance of an event occurring can be described as being from certain (a probability of 1) to impossible (a probability of 0).
• Terms used to describe the chance of an event occurring include improbable, unlikely, fifty-fifty, likely and probable.
• The chance of an event occurring can be described by counting the possible outcomes and sometimes by relying on our general knowledge.
Single event probability
• The probability of an event can be found using the formula
• Probabilities are often written as fractions, but can also be expressed as decimals or percentages.
Relative frequency
• Relative frequency describes how often an event has occurred.
• It is found by dividing the number of times an event has occurred by the total number of trials.
Random numbers and simulations
• Computers and calculators can generate random decimals between 0 and 1. • To generate 10 random numbers between 1 and 6 inclusive use the randint(1, 6, 10)
function on the TI-83 graphics calculator, or randbetween(1, 6) on an Excel spreadsheet.
Long-run proportion
• Long-run proportion is when the experimental probability obtained using random numbers gets closer to the true probability.
• The probability of an event, A, is symbolised by Pr(A).
s
ummary
1 Graham and Marcia are playing a game. To see who starts they each take a card from a standard deck. The player with the highest card starts. Graham takes a five. Describe Marcia’s chance of taking a higher card.
2 Describe each of the following events as being either certain, probable, even chance (fifty-fifty), unlikely or impossible.
a Rolling a die and getting a number less than 6
b Choosing the eleven of diamonds from a standard deck of cards
c Tossing a coin and it landing Tails
d Rolling two dice and getting a total of 12
e Winning the lottery with one ticket
3 Give an example of an event which is:
a certain b impossible.
4 The Chen family are going on holidays to Alice Springs during January. Are they more likely to experience hot weather or cold weather?
5 List each of the events below in order from most likely to least likely.
Winning a lottery with 1 ticket out of 100 000 tickets sold Rolling a die and getting a number greater than 1
Selecting a blue marble out of a bag containing 14 blue, 15 red and 21 green marbles Selecting a picture card from a standard deck
6 Mark and Lleyton are tennis players who have played eight previous matches. Mark has won six of these matches. When they play their ninth match, who is more likely to win? Explain your answer.
7 The numbers 1 to 5 are written on the back of 5 cards that are turned face down. Michelle then chooses one card at random. Michelle wants to choose a number greater than 2. List the sample space and all favourable outcomes.
8 A barrel contains 25 numbered balls. One ball is drawn from the barrel. Find the probability that the marble drawn is:
9 A card is to be chosen from a standard deck. Find the probability that the card chosen is:
a 13 b 7 c an odd number
d a square number e a prime number f a double-digit number.
a the 2 of clubs b any 2 c any club
d a black card e a court card f a spot card.
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10 A video collection has 12 dramas, 14 comedies, 4 horror and 10 romance movies. If I choose a movie at random from the collection, find the probability that the movie chosen is:
11 The digits 5, 7, 8 and 9 are written on cards. They are then arranged to form a four-digit number. Find the probability that the number formed is:
12 A raffle has 2000 tickets sold and has two prizes. Michelle buys five tickets.
a Find the probability that Michelle wins 1st prize.
b If Michelle wins 1st prize, what is the probability that she also wins 2nd prize?
13 From every 100 televisions on a production line, two are found to be defective. If you choose a television at random, find the relative frequency of defective televisions.
14 It is found that 150 of every thousand 17-year-old drivers will be involved in an accident within one year of having their driver’s licence.
a What is the relative frequency of a 17-year-old driver having an accident?
b If the average cost to an insurance company of each accident is $5000, what would be the minimum premium that an insurance company should charge a 17-year-old driver?
15 What could you enter on a TI-83 graphics calculator to generate 10 random integers between −20 and +20?
16 A dice game is played whereby a die is rolled. It costs $1 to play. If the player rolls a 6 they receive a return of $6 (they actually win $5 as it cost $1 to play). If the player rolls a 1 they get the $1 cost of the game back. Any other number means they lose $1. Use a long-run proportion to calculate whether the player can expect to win or lose money.
a a comedy b a horror c not romance.
a 7895 b odd c divisible by 5
d greater than 7000 e less than 8000.
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