1 A random sampling of 80 ceramic tiles produced at a ceramics factory reveals 8 scratched, 3 chipped and 4 broken tiles. Estimate the probability that a tile produced at the factory will be:
a scratched
b chipped or broken c damaged in some way.
2 A card is randomly selected from a deck, its suit is noted, and then the card is placed back in the deck.
The experiment is repeated to obtain a second card.
a List the possible outcomes for selecting 2 cards in this way.
b Are the outcomes all equally likely? Explain.
3 A game show host spins the wheel shown at right. What is the probability that the wheel ends on:
a the jackpot ($1000 prize)?
b a prize greater than $50?
4 A letter is chosen at random from each of the words GO BLUES. Represent all possible outcomes on a tree diagram and find the probability that:
a G and B are chosen b S is chosen c G or S is chosen.
5 A standard die is thrown and the spinner shown at right is spun.
a Show all possible outcomes on a lattice diagram.
b Find the probability of getting a number greater than 4 on the die and an odd number on the spinner.
6 A set of 20 uniformly sized cards numbered 1 to 20 is shuffled. What is the probability of drawing a number less than 8 or an even number from this set?
7 A class of 30 students was asked if there was a pet dog at home and if the students were responsible for pooper scooping before the backyard lawn was mown. Fourteen students had a dog but only 6 did the pooper scooping.
a Draw a Karnaugh map showing this information.
b Complete a probability table.
c State the probability that a randomly selected student has a dog but avoids pooper scooping.
8 If Pr(A) = 0.3 and Pr(B | A) = 0.4, find Pr(A ∩ B).
9 Two identical, equally accessible cookie jars sit on a kitchen bench. Jar 1 contains 6 chocolate and 9 plain biscuits, and jar 2 contains 12 chocolate and 8 plain biscuits. One biscuit is selected randomly from one of the jars. If a chocolate biscuit is selected, what is the probability that it came from jar 1?
10 Of 50 people surveyed, 35 played tennis and 26 played netball. Everyone surveyed played at least one of these sports.
a How many people played both netball and tennis?
b If one person is selected at random, what is the probability that:
i he/she plays tennis only?
ii he/she plays netball?
iii he/she plays tennis, given that he/she also plays netball?
mUlTiplE
ChoiCE 1 Twelve nuts are taken from a jar containing macadamias and cashews. If 3 macadamias are obtained, the estimated probability of obtaining a cashew is:
a 1
12 B 1
4 C 1
3 d 3
4 E 3
1
2 From a normal pack of 52 playing cards, one card is randomly drawn and replaced. If this is done 208 times, the number of red or picture cards (J, Q or K) expected to turn up is:
a 150 B 130 C 120 d 160 E 128
3 A cubic die with faces numbered 2, 3, 4, 5, 6 and 6 is rolled. The probability of rolling an even number is:
a 13 B 23 C 16 d 56 E 12
ShorT anSWEr
$1000 Jackpot $60
$100Lose
$100
Boody Prize
$40
$30 $500
$50
$200
Loseit all
$10
2 3
1
4 The probability of rolling an odd number or a multiple of 2 using the die in question 3 is:
a 1 B 13 C 14 d 34 E 23
5 The tree diagram that describes the outcomes when three coins are tossed is:
a Bag B contains 3 yellow and 2 green marbles, as shown at right. A marble is drawn from Bag A, then one is taken from Bag B. Which diagram below best illustrates this situation?
a Yellow RY
7 If Pr(A) = 0.6, Pr(B) = 0.7 and Pr(A ∪ B) = 0.8, then Pr(A ∩ B) is:
a 0.1
B 0.5
C 0.9
d 0.2
E 0.6
8 Consider the Karnaugh map at right, where C is ‘people who like comedy movies’ and A is ‘people who like action movies’.
The number of people who like only action movies is:
a 26 B 30 C 18
d 20 E 23
9 Which of the following alternatives gives the correct values of a and b in the probability table at right?
a a = 0.2, b = 0.3
B a = 0.3, b = 0.2
C a = 0.3, b = 0.6
d a = 0.6, b = 0.3
E a = 0.7, b = 0.4
10 If Pr(B | A) = 0.45 and Pr(A ∩ B) = 0.35, then Pr(A) is:
a 79 B 38 C 49 d 58 E 89
11 A fair coin is tossed twice. If A = Tails on first toss, B = Heads on second toss and C = both tosses are Heads, which of the following is true?
a Pr(A) = 14
B Pr(B) = Pr(C)
C A and B are independent.
d A and C are independent.
E B and C are independent.
12 An archer has a probability of approximately 3
8 of hitting the bullseye from a particular distance. The spinner at right is used to simulate 10 rounds of 4 shots at such a target, and the
results are as follows, where B = blue (bullseye) and R = red (non-bullseye).
B, R, R, R B, R, B, R R, B, R, R R, B, R, R R, B, B, R
R, R, B, B R, B, R, R R, R, R, R R, B, B, R B, R, B, B
Based on this simulation, the probability of getting 2 bullseyes in a round of 4 shots is:
a 14 B 38 C 25 d 58 E 34
ExTEndEd
rESponSE 1 For a transition matrix T = 0.22 0.33 0.78 0.67
and an initial state matrix S0 = 0.5 0.5
, calculate, accurate to 3 decimal places:
a S1 b S4.
2 The manager of a snow resort has noticed that, if it snows on a given day, there is a 70% chance that it will snow the following day. If it does not snow, there is only a 30% chance that it will snow the following day. John arrived on a Saturday when the weather was sunny and clear.
a What is the probability that he will have fresh snow the following Tuesday?
b What is the probability that he will have fi ne weather for the drive home on the following Saturday? (Give answers to 3 decimal places.)
3 A factory has a machine in poor working condition that often produces faulty components. If it produces a faulty component, there is a probability of 0.25 that it will follow this up with another faulty component. However, each time it produces a good component, there is a probability of only 0.05 that
A A′
C 19
C′ 25
26 50
B B′
A 0.4 0.7
A′ a b
0.4
it will next produce a faulty component. If there is a 20% chance that the first component of the day is faulty, set up the initial state matrix and find the probability that:
a the second component is faulty b the fi fth component is faulty.
4 One student is selected at random from each of Years 7, 8 and 9. If there are 148 girls and 114 boys in Year 7, 126 girls and 97 boys in Year 8, and 115 girls and 122 boys in Year 9, find the probability that all students chosen are boys. Give your answer to 3 decimal places.
5 The probability that the newspaper is delivered to Geoff’s house before 8.00 am is 5
7, and the probability that he arrives at work to fi nd a free parking space in his company’s car park is 9
10. Assuming these events are independent:
a use a CAS calculator to conduct a simulation of 28 days’ duration
b fi nd the probability, based on the simulation, that on one day Geoff misses out on a company car park space and his paper arrives late
c calculate the theoretical probability of the combination of events described in part b.
6 Consider 3 fair coins being tossed.
If A = Heads with the fi rst coin, B = Tails with the second coin and C = Tails with the third coin:
a list the event space (for example, use HTH for
‘Head then Tail then Head’) b fi nd:
i Pr(A) ii Pr(B) iii Pr(C) iv Pr(A ∩ B) v Pr(A ∩ C) vi Pr(B ∩ C) vii Pr(A ∩ B ∩ C) viii Pr(A) × Pr(B) × Pr(C)
c propose how you might defi ne independence for 3 events.
7 A large number of asthma sufferers were asked to volunteer for the testing of a new drug. Only some of the volunteers were given the drug, but all of the volunteers were observed to see if they developed asthma on a smoggy day. The results are shown in the table below.
Given drug Not given drug
Developed asthma 148 59
Did not develop
asthma 566 184
a How many people were selected to take part in the study?
b If a person was randomly selected from the volunteers, what is the probability that they were given the drug?
c What is the probability that a randomly selected person developed asthma on this day?
d Given that a volunteer was given the drug, what is the probability that they developed asthma?
e From this information, what can you conclude about the effectiveness of the drug in preventing asthma?
It was decided that conclusive observations about the effectiveness of the drug could not be made after one day, so the same volunteers continued with the study for three months. (Assume that the number of people given the drug is unchanged.) The results were:
395 people were given the drug and did not have an asthma attack
143 people were given the drug but had exactly one episode of asthma per month 97 people were not given the drug and developed asthma more than once a month 84 people were not given the drug and had exactly one episode of asthma each month.
f Represent this information using a Venn diagram.
g How many people were given the drug and had more than one episode of asthma per month?
h How many people in the study were not given the drug and did not have an episode of asthma?
i Given that a volunteer had been given the drug, what is the probability that they have more than one episode of asthma per month?
j Given that a volunteer had more than one episode of asthma per month, what is the probability they had taken the drug?
diGiTal doC doc-9812 Test Yourself Chapter 11