Answered: Suppose that the genders of the three…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Suppose that the genders of the three children of a family are soon to be revealed. An outcome is represented by a string of the sort

GBB

(meaning the oldest child is a girl, the second oldest is a boy, and the youngest is a boy).

The

outcomes are listed below. Assume that each outcome has the same probability.

Complete the following. Write your answers as fractions.

(a)Check the outcomes for each of the three events below. Then, enter the probability of each event.

	Outcomes	Probability
	BBB	BBG	BGB	BGG	GBB	GBG	GGB	GGG
Event X: The last child is a boy
Event Y: Exactly one child is a girl
Event X and Y: The last child is a boy and exactly one child is a girl

(b)Suppose exactly one child is a girl. (That is, Event

occurs.) This will limit the possible outcomes. From the remaining outcomes, check the outcomes for Event

. Then, enter the probability that Event

occurs given that Event

occurs.

	Outcomes given exactly one child is a girl	Probability
		BBG	BGB	GBB
Event X: The last child is a boy

(c)Give the following probabilities and select the correct option below.

PX and YPY	=
PX\|Y	=
PX and YPY	▼?	PX\|Y

EXPLANATION

(a)We are asked to find the probabilities of the three given events.

Since all

outcomes have the same probability, each outcome has probability

To compute the probability of an event, we just add the probabilities of the outcomes making up the event.

Event

X

is "The last child is a boy". The outcomes in that event are

BBB

,

BGB

,

GBB

, and

GGB

.
The probability of the event is thus

=4812

.
Event

Y

is "Exactly one child is a girl". The outcomes in that event are

BBG

,

BGB

, and

GBB

.
The probability of the event is thus

38

.
Event

X

and

Y

is "The last child is a boy and exactly one child is a girl". The outcomes in that event are

BGB

and

GBB

.
The probability of the event is thus

=2814

.

	Outcomes	Probability
	BBB	BBG	BGB	BGG	GBB	GBG	GGB	GGG
Event X: The last child is a boy									12
Event Y: Exactly one child is a girl									38
Event X and Y: The last child is a boy and exactly one child is a girl									14

(b)We are asked to find the probability that the last child is a boy given exactly one child is a girl.

If we know exactly one child is a girl, then there are only

possible outcomes.

Since all

outcomes have the same probability, each outcome has probability

.
Two of the possible outcomes have a boy as the last child:

BGB

and

GBB

.
The probability of the event is thus

	Outcomes given exactly one child is a girl	Probability
		BBG	BGB	GBB
Event X: The last child is a boy					23

(c)We are asked to find

PX and YPY

.
To do this, we can use the probabilities found in part (a).

Note that

PX and Y

means the probability that "the last child is a boy and exactly one child is a girl".
From part (a), we have

=PX and Y14

And

means the probability that "exactly one child is a girl".
From part (a), we have

=PY38

.
So we get the following.

PX and YPY	=1438
	=·1483
=23

We are asked to find

PX|Y

, the conditional probability of event

given event

.
In other words, we must find the probability that the last child is a boy given exactly one child is a girl.

From part (b), we get the following.

=PX|Y23

Note that

PX and YPY

and

PX|Y

both equal

So we get the following.

PX and YPY=PX|Y

It turns out that this equation is true for any two events

and

for which

>PY0

.
This equation is the formula for computing conditional probability.

ANSWER

(a)Check the outcomes for each of the three events below. Then, enter the probability of each event.

	Outcomes	Probability
	BBB	BBG	BGB	BGG	GBB	GBG	GGB	GGG
Event X: The last child is a boy									12
Event Y: Exactly one child is a girl									38
Event X and Y: The last child is a boy and exactly one child is a girl									14