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A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 3: Q.3.24 (page 99)

Each of $2$ balls is painted either black or gold and then placed in an urn. Suppose that each ball is colored black with probability $\frac{1}{2}$ and that these events are independent.
(a) Suppose that you obtain information that the gold paint has been used (and thus at least one of the balls is painted gold). Compute the conditional probability that both balls are painted gold.
(b) Suppose now that the urn tips over and 1 ball falls out. It is painted gold. What is the probability that both balls are gold in this case? Explain.

Short Answer

Expert verified

a)The conditional probability that both balls are painted gold $\frac{1}{3} [2 e x]$

b) The probability that both balls are gold is $\frac{1}{2}$

Step by step solution

01

Step1:Introduction

Considered events:

$G_{1}$ -The first of the two balls is gold in color.

$G_{2}$ - The second of the two balls is gold in color.

Probabilities

$P (G_{1}) = \frac{1}{2} 鈬� P (G_{1}^{c}) = \frac{1}{2}$

$P (G_{2}) = \frac{1}{2} 鈬� P (G_{2}^{c}) = \frac{1}{2}$

The events $G_{1}$ and $G_{2}$ are independent

02

FindPG1G2∣G1∪G2

Begin by defining conditional probability.

$P (G_{1} G_{2} 鈭� (G_{1} 鈭� G_{2})) = \frac{P [G_{1} G_{2} (G_{1} 鈭� G_{2})]}{P (G_{1} 鈭� G_{2})}$

We use this to change the definition (1),

localid="1649413957621" $P [G_{1} G_{2} 鈭� (G_{1} 鈭� G_{2})] = \frac{P (G_{1} G_{2})}{P [{(G_{1}^{c} G_{2}^{c})}^{c}]}$

$= \frac{P (G_{1}) P (G_{2})}{1 鈭� P [(G_{1}^{c} G_{2}^{c})]}$

$= \frac{P (G_{1}) P (G_{2})}{1 鈭� P (G_{1}^{c}) P (G_{2}^{c})}$

$= \frac{\frac{1}{2} 脳 \frac{1}{2}}{1 鈭� \frac{1}{2} 脳 \frac{1}{2}}$

$= \frac{1}{3}$

03

Step2:The probability that the other one is golden(part b)

What are the chances that the other ball will be golden as well?

Either $G_{1}$ happens, and the desired probability is $P (G_{2} 鈭� G_{1})$

or $G_{2}$ happens, and the desired probability is $P (G_{1} 鈭� G_{2})$ .

According to the definition of independence,

$\begin{aligned} P (G_{1} 鈭� G_{2}) = P (G_{1}) = \frac{1}{2} \\ P (G_{2} 鈭� G_{1}) = P (G_{2}) = \frac{1}{2} \end{aligned}$

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Most popular questions from this chapter

Consider an urn containing $12$ balls, which $8$ are white. A sample of size $4$ is to be drawn with replacement (without replacement). What is the conditional probability (in each case) that the first and third balls drawn will be white given that the sample drawn contains exactly $3$ white balls?

(a) Prove that if $E$ and $F$ are mutually exclusive, then

localid="1647926638131" $P (E 鈭� E 鈭� F) = \frac{P (E)}{P (E) + P (F)}$

(b) Prove that if localid="1647926673038" $E_{i}, i 鈮� 1$ are mutually exclusive, then

localid="1648539605315" $P (E_{j} 鈭� {鈭�}_{i = 1}^{鈭�} E_{i}) = \frac{P (E_{j})}{{鈭�}_{i = 1}^{鈭�} P (E_{i})}$

A total of 48 percent of the women and 37 percent of the men who took a certain 鈥渜uit smoking鈥� class remained nonsmokers for at least one year after completing the class. These people then attended a success party at the end of a year. If 62 percent of the original class was male,

(a) what percentage of those attending the party were women?

(b) what percentage of the original class attended the party?

If you had to construct a mathematical model for events E and F, as described in parts (a) through (e), would you assume that they were independent events? Explain your reasoning.

(a) E is the event that a businesswoman has blue eyes, and F is the event that her secretary has blue eyes.

(b) E is the event that a professor owns a car, and F is the event that he is listed in the telephone book.

(c) E is the event that a man is under 6 feet tall, and F is the event that he weighs more than 200 pounds.

(d) E is the event that a woman lives in the United States, and F is the event that she lives in the Western Hemisphere.

(e) E is the event that it will rain tomorrow, and F is the event that it will rain the day after tomorrow.

What is the probability that at least one of a pair of fair dice lands on 6, given the sum of the dice $i$ $i, i = 2, 3, . . ., 12$ ?

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