2 5 ---4
12--- 16
20--- 24
50
---Experimental probability formula:
The experimental probability= number of times this event occurred
---More and more statistics are being collected (empirical evidence) from which predictions can be made. Probabilities based on this evidence are used to determine the cost of insurance, life expectancy and the likelihood of events occurring. These estimates are often called empirical probabilities and are a type of experimental probability. If Australia had beaten England at the SCG four of the last five times they have played there, then it would be highly likely that Australia will win next time.
Experimental probabilities are usually based on an examination of a sample or trial run of the activity under examination.
The first 100 vehicles to pass a checkpoint gave the results in the table. If these figures truly represent the traffic at any time past this checkpoint, determine the experimental probability that the next vehicle will be:
a a car b a motor cycle
c a bus d not a car
e not a car or truck
Worked examples
1 A farmer collects 10 eggs and finds that 2 of them are bad.
If he chose another egg what is the chance of getting another bad one?
2 The contents of 20 matchboxes were examined and the results recorded.
If the contents of a similar box of matches were counted, what would be the experimental probability that it would contain 50 matches or more?
Solutions
1 Since 2 of the first 10 eggs were bad, it seems that , or of the farmer’s eggs might be bad.
So, if the first 10 eggs were truly representative of all the farmer’s eggs, then the chance of picking another bad one is , or 1 out of 5.
2 In the sample, 14 of the 20 boxes had more than 50 matches.
Experimental probability =
=
In boxes like these we would expect the chance of choosing one with 50 or more matches to be or .
Number of matches 48 49 50 51 52 53
Number of boxes 1 5 8 3 2 1
■ The probability of an event occurring in an experiment is the same as its ‘relative frequency’.
2
---number of times this event occurred total number in sample
---Exercise 4:02
Experimental probability NS5·1·3 1 In a class of 24, 12 were blonde, 10 werebrunette, 2 were redheads. Find the probability that one student chosen at random will be:
a blonde b not brunette Foundation Worksheet 4:02
1
Type of vehicle Frequency
Cars 70
Trucks 15
Motor cycles 10
Buses 5
We examined the contents of 37 packets of coloured lollies. The average number of each colour in a packet is shown in the table. One lolly is taken at random from a new packet. Use these results to determine:
a which colour is most likely to be picked
b which colour has the least chance of being picked c the probability that it is red
A factory tested a sample of 100 light bulbs and 5 were found to be faulty. From these results, what is the probability of buying a faulty light bulb? What is the probability of buying a good bulb?
Sid Fowler recorded the number of eggs his chickens laid each day, for 6 weeks. The results are shown in the table.
If these results are typical for Sid’s chickens at any time of the year find, as a fraction, the probability that on any particular day the number of eggs laid is:
a 2 b 4
c 2 or more
Convert each of your answers to a percentage correct to the nearest whole per cent.
A survey of 100 households was taken to determine how many used certain washing powders. Based on these results, what is the probability of a household chosen at random:
a using Foam brand?
b using Supersoap?
c not using Pow?
d not using any of these four brands?
Jenny tossed four coins 30 times and the number of heads was recorded each time. The histogram shows the results.
a From this experiment what is the probability that when four coins are thrown there will be:
i no heads?
ii two heads?
iii at least three heads?
b If this experiment were to be repeated, would you expect the same results?
Colour Frequency
Number of heads 6
A dice was thrown 50 times and the results were recorded.
Using the results in the table:
a What is the experimental probability of throwing:
i a six?
ii a two?
iii an odd number?
b Would you expect to get seven ones every time a dice is thrown 50 times?
c If the dice was thrown 500 times what pattern of results would you expect?
of Year 9 students have a shoe size greater than 10 , but have a shoe size less than 11 . What is the chance of a Year 9 student having a shoe size:
a less than or equal to 10 ? b 11 or larger?
c between 10 and 11 ?
(Note: This is every size not in a and b.)
a A company makes electrical equipment and tests its products regularly to see if they will survive the products warranty period. In a test of 1000 articles, it finds 73 are faulty in some way. Based on this evidence, if they sell 60 000 articles, how many will they expect to fail during the warranty period?
b A bag contains a total of 1000 red and blue marbles. Adelaide chooses 40 marbles at random from the bag and finds that 28 are red and 12 are blue. On the basis of her results, estimate how many red marbles are in the bag.
A biologist wishes to estimate the number of fish in a dam. He catches 100 fish, tags them and then releases them back into the dam. He assumes that the chance of recatching any of the tagged fish will depend on the size of the fish population. The larger the population, the smaller the chance there would be of recatching any of the fish. He returns in two weeks and this time he catches 200 fish of which 6 are tagged. Based on his assumption, he concludes that the chance of catching a tagged fish is 3%. How can he then estimate the number of fish in the dam?
Kate, Paul and Jason have been asked to play a game in which three dice are rolled. The person rolling the dice wins if a six appears on any of the dice.
Number shown Frequency
1 7
2 5
3 5
4 10
5 9
6 14
7
8 4
20--- 1
2 ---19
20--- 1
2 ---1
2 ---1
2
---1
2--- 1
2
---9
10
11
They decided to do a simulation to estimate their winning chances. The results were as follows:
a Use each person’s results to calculate the chance of winning.
b Is it true that Paul’s results would be the most unreliable because he performed the experiment fewer times than Jason and Kate?
c Is it true that Kate’s results would give the best estimate because she performed the experiment more times than the other two?
d Do you think it is true that the results become more reliable as the number of experimental trials increases?
e How could the results above be used to give a more reliable estimate of the chance of winning? What would it be?
No sixes One six Two sixes Three sixes
Kate 45 33 2 0
Paul 12 8 0 0
Jason 18 20 2 0
When tossing a coin we assume that the probability of getting a head is or 50%, but is this true?
Luke tossed a coin five times and graphed the percentage of heads after each toss.
He tossed: head, head, tail, head, tail.
His graph is shown below.
1 Toss a coin ten times and graph the percentage of heads after each toss. Did the percentage get closer to 50% as the number of tosses increased?
2 Would this experiment be a reasonable simulation for the gender of babies born in a local hospital?
3 If you repeat this investigation, would you obtain the same graph? In what way would the second graph resemble the first?