3. What is the smallest number of people needed
before it is certain that at least two were born
in the same month? Why?
4. Our goal is to obtain a better estimate of the
probability that in a group of 25 people, at
least two were born on the same day of the
year.
a. What would you guess this probability to
be?
b. Perform a total of at least 25 trials to add to
the results shown in Table 5.13. (This will
be somewhat time-consuming by hand.)
What is your estimate of the probability?
c. Was your guess in part (a) close to your
estimate in part (b)?
5. Determine the experimental probability that
at least two people in random groups of the
following sizes have the same birthday:
a. 5
b. 10
c. 15
d. 20
(Your accuracy depends on your stamina, on
obtaining the assistance of other class members, or on the use of a computer simulation
program.)
6. Draw a graph of the data of Exercise 5. Let
the x value be size of the group, and let the
y value be the experimental probability of a
shared birthday.
7. Suppose each of three people all select a number between 1 and 10. What is the probability
that at least two of them choose the same
number?
8. Suppose each of six students selects at random
an integer between 1 and 52, inclusive.
a. What is the probability that at least two of
the students will select the same integer?
(You may wish to use a box model instead
of a table of random digits.)
b. Translate this into a birth week problem,
assuming a year contains exactly 52 weeks.
For additional exercises, see page 724.
5.9 CONDITIONAL PROBABILITY
Suppose we wish to find a probability when certain information is given or
some event is known to have happened. We denote the probability that an
event A occurs given that B is true or is known to have occurred by P(A 兩 B)
or p(A 兩 B), and we read this as “the probability of A given B.” It is also
often referred to as the conditional probability of A given B. As usual P
denotes an experimental probability and p denotes a theoretical probability.
We will discover in Chapter 14 that the concept of conditional probability is
an important one and has a close relationship to independence. But for now
we merely introduce the idea.
Example 5.14
A carnival spieler shows an audience three cards. One card is red on both sides,
one blue on both sides, and one red on one side and blue on the other. Someone in
the audience shuffles the cards and places one on the table in such a way that no
one knows what color is on the bottom side. The top side is red. The spieler says,
“Obviously this is not the blue-blue card. Then it is either the red-red card or the
red-blue card.” Suppose the spieler will bet even money that it is the red-red card
(he will pay you $1 if he is wrong and you will pay him $1 if he is right). Is this a fair
bet? (From W. L. Ayres, C. G. Fry, and H. F. S. Jonah, General College Mathematics,
3rd ed., McGraw-Hill, 1970, p. 234.)
Solution
Since any of the three cards would be put on the table with equal probability,
we will use a die to simulate this problem. We set up the following step-by-step
procedure.
1. Choice of a Model: There are six possible ways of choosing and laying a card
on the table, because each of the three possibly chosen cards has two faces to choose
as the upward face. Let a toss of 1 or 2 represent the red-red card being put on the
table, 3 or 4 represent the blue-blue card, and 5 or 6 the red-blue card. Further, if
the toss is 5, let us say that the red side is up, and if the toss is a 6 we will say that
the blue side is up.
2. Definition of a Trial: Toss the die. Since we are given that the top side is red,
we will ignore a toss of 3, 4, or 6, which correspond to a blue side being up. Toss until
a 1, 2, or 5 appears. This is one trial of the experiment. It amounts to conditioning
on the red side being up.
3. Definition of a Successful Trial: Call the event that the toss is a 1 or 2 a
success, because this would correspond to the red-red card being placed on the
table.
4. Repetition of Trials: Do at least 100 trials.
The results listed in Table 5.14 were obtained for 10 trials.
5. Finding the Probability of a Successful Trial: Estimate p(red-red card 兩
red face up) as the number of successes divided by the number of trials, with all
trials that did not produce a red-up card intentionally eliminated. Thus we estimate
that
P(red-red card 兩 red face up) ⳱
7
⳱ 0.7
10
Of course, 10 trials is not enough to produce much accuracy. Interestingly,
p(red-red card 兩 red face up) ⳱ 2/3, in contrast with the intuitive (and incorrect)
notion that p(red-red card 兩 red face up) ⳱ 1/2. If we ran 400 trials (easy to do with
Table 5.14
Estimating p (red-red card 兩 red face up)
Trial
Die tosses
Success?
1
2
3
4
5
6
7
8
9
10
2
4, 2
3, 4, 1
5
6, 2
1
2
3, 5
6, 5
2
Yes
Yes
Yes
No
Yes
Yes
Yes
No
No
Yes
a computer), we would expect to be quite close to the true probability, and close
enough to easily decide that 1/2 is not correct, as the spieler in effect claimed with
his even-money bet.
Another such puzzling example is the Let’s Make a Deal television show
problem. Behind each of two curtains is a pig, and behind the third is a
BMW car. You choose a curtain at random (clearly, p(win car) ⳱ 13 ). But the
announcer reveals a pig behind one of the other curtains before you get to
look behind your curtain. He then says that if you wish, you can change the
curtain you have chosen. Should you?
Most people would think that changing curtains does not alter your
chances of winning. But what is p(win 兩 change curtain)? This problem
yields the startling result that
p(win 兩 change curtain) ⳱
2
3
while
p(win 兩 do not change curtain) ⳱
1
3
Thus you should change curtains!
Do you believe this? If not, you could use the five-step method to test it.
SECTION 5.9 EXERCISES
1. Continue the problem of the carnival spieler
by doing at least another 30 trials and obtaining a new estimate of the probability that the
spieler wins the bet.
2. What is the experimental probability that a
family with three children has all boys if you
know that at least one of the children is a boy?
3. Sometimes a conditional probability can be
found theoretically by simple logical reasoning. Out of four bottles of milk two are spoiled.
John buys milk and then Jack buys milk. Find
a. p(Jack gets spoiled milk 兩 John got spoiled
milk)
b. p(Jack gets spoiled milk 兩 John got fresh
milk)
4. A somewhat absent-minded hiker forgets to
bring his insect repellent on 30% of his hikes.
The probability of being bitten is 90% if he
forgets the repellent, and 20% if he uses the
repellent. Perform at least 50 trials simulating
whether the hiker is bitten or not.
a. What is the experimental probability that
he will be bitten?
b. Now look only at the times when the hiker
was bitten. What is the total number of
times he was bitten? Of these times, how
many times did he forget to bring his repellent?
c. Using the information from part (b), calculate the experimental probability that he
forgot his repellent given that he was bitten.