HONG KONG CHINESE WOMEN'S CLUB COLLEGE
F.7 APPLIED MATHEMATICS
SUPPLEMENTARY EXERCISE
Selected Examples on Probability
CLASSICAL DEFINITION OF PROBABILITY
1. A card is drawn at random from an ordinary pack of 52 playing cards. Find the probability
that the card (a) is seven, (b) is not a seven.
2. Compare the probabilities of scoring a 4 with one die and a total of 8 with two dice.
3. Two fair coins are tossed. Illustrate the possible outcomes on a possibility space diagram and
find the probability that two heads are obtained.
4. A coin and a die are thrown together. Draw a possibility space diagram and find the
probability of obtaining (a) a head, (b) a number greater than 4, (c) a head and a number
greater than 4, (d) a head or a number greater than 4.
5. Events A and B are such that P(A)=19/30, P(B)=2/5 and P(AB)=4/5. Find P(AB).
6. In a group of 20 adults, 4 out of the 7 women and 2 out of the 13 men wear glasses. What is
the probability that a person chosen at random from the group is a woman or someone who
wears glasses?
MUTUALLY EXCLUSIVE EVENTS
1. In a race the probability that John wins is 0.3, the probability that Paul wins is 0.2 and the
probability that Mark wins is 0.4. Find the probability that (a) John or Mark wins, (b) neither
John nor Paul wins. Assume that there are no dead heats.
2. A card is drawn at random from an ordinary pack of 52 playing cards. Find the probability
that the card is (a) a club or a diamond, (b) a club or a king.
CONDITIONAL PROBABILITY
1. Given that a heart is picked at random from a pack of 52 playing cards, find the probability
that it is a picture card.
2. When a die is thrown, an odd number occurs. What is the probability that the number is
Page 1
prime?
3. Two tetrahedral dice, with faces labeled 1, 2, 3 and 4, are thrown and the number on which
each lands is noted. The core’ is the sum of these two numbers. Find the probability that
(a) the score is even, given that at least one die lands on a 3, (b) at least one die lands on a 3,
given that the score is even.
4. A bag contains 10 counters, of which 7 are green and 3 are white. A counter is picked at
random from the bag and its color is noted. The counter is not replaced. A second counter
is then picked out.
Find the probability that (a) the first counter is green, (b) the first counter is green and the
second counter is white, (c) the counters are of different colors.
INDEPENDENT EVENTS
1. A die is thrown twice. Find the probability of obtaining a 4 on the first throw and an odd
number on the second throw.
2. A bag contains 5 red counters and 7 black counters. A counter is drawn from the bag, the
color is noted and the counter is replaced. A second counter is then drawn. Find the
probability that the first counter is red and the second counter is black.
3. A fair die is thrown twice. Find the probability that (a) neither throw results in a 4, (b) at least
one throw results in a 4.
4. Events A and B are such that P(A)=1/3 and P(AB)=1/12. If A and B are independent events,
find (a) P(B), (b) P(AB).
5. Two events A and B are such that P(A)=1/4, P(A冰)=1/2 and P(B再)=2/3.
6. The probability that a certain type of machine will break down in the first month of operation
is 0.1. If a firm has two such machines which are installed at the same time, find the
probability that, at the end of the first month, just one has broken down.
MORE THAN TWO EVENTS
1. Records in a music shop are classed in the following sections: classical, popular, rock, folk and
jazz. The respective probabilities that a customer buying a record will choose from each
section are 0.3, 0.4, 0.2, 0.05 and 0.05. Find the probability that a person (a) will choose a
record from the classical or the folk or the jazz sections, (b) will not choose a record from the
rock or folk or classical sections.
Page 2
2. A die is thrown four times. Find the probability that a 5 is obtained each time.
3. Three people in an office decide to enter a marathon race. The respective probabilities that
they will complete the marathon are 0.9, 0.7 and 0.6. Find the probability that at least two will
complete the marathon. Assume that the performance of each is independent of the
performances of the others.
PROBABILITY TREE
1. A bag contains 8 white counters and 3 black counters. Two counters are drawn, one after each
other. Find the probability of drawing one white and one black counter, in any order, (a) if the
first counter is replaced, (b) if the first counter is not replaced.
2. The probability that a golfer hits the ball on to the green if it is windy as he strikes the ball is
0.4, and the corresponding probability if it is not windy as he strikes the ball is 0.7. The
probability that the wind will blow as he strikes the ball is 0.3.
Find the probability that (a) he hits the ball on to the green, (b) it was not windy, given that he
does not hit the ball on to the green.
3. Events A and B are such that P(A)=1/3, P(B再)=1/4 and P(A冰)=4/5.
By drawing a tree diagram, or otherwise, find (a) P(B再), (b) P(AB), (c) P(B), (d) P(AB).
4. A fair coin is tossed three times. What is the probability of obtaining (a) exactly two heads,
(b) at least two heads?
BAYES’ THEOREM
1. Three girls, Aileen, Barbara and Cathy, pack biscuits in a factory. From the batch allotted to
them Aileen packs 55%, Barbara 30% and Cathy 15%. The probability that Aileen breaks
some biscuits in a packet is 0.7, and the respective probabilities for Barbara and Cathy are 0.2
and 0.1. What is the probability that a packet with broken biscuits found by the checker was
packed by Aileen?
2. Three children, Catherine, Michael and David, have equal plots in a circular patch of garden.
The boundaries are marked out by pebbles. Catherine has 80 red and 20 white flowers in her
batch, Michael has 30 red and 40 white flowers and David has 10 red and 60 white flowers.
Their young sister, Mary, wants to pick a flower for her teacher.
(a) Find the probability that she picks a red flower if she chooses a flower at random from the
garden, ignoring the boundaries.
(b) Find the probability that she picks a red flower if she first chooses a plot at random.
Page 3
(c) If she picks a red flower by the method described in (b), find the probability that it comes
from Michael's plot.
PROBLEMS INVOLVING AN 'AT LEAST' SITUATION
1. (a) Find the probability of obtaining at least one 6 when 5 dice are thrown.
(b) Find the probability of obtaining at least one 6 when n dice are thrown.
(c) How many dice must be thrown so that the probability of obtaining at least one 6 is at
least 0.99?
PROBLEMS INVOLVING THE USE OF AN INFINITE GEOMETRIC PROGRESSION
1. A,B and C, in that order, throw a tetrahedral die. The first one to throw a 4 wins. The game is
continued indefinitely until someone wins. Find the probability that (a) A wins, (b) B wins,
(c) C wins.
Page 4