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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.3.38 (page 100)

Urn $A$ has $5$ white and $7$ black balls. Urn $B$ has $3$ white and $12$ black balls. We flip a fair coin. If the outcome is heads, then a ball from urn $A$ is selected, whereas if the outcome is tails, then a ball from urn $B$ is selected. Suppose that a white ball is selected. What is the probability that the coin landed tails?

Short Answer

Expert verified

The probability that the coin landed tails is $\frac{12}{37}$ .

Step by step solution

01

Given information

Urn $A$ has $5$ white and $7$ black balls. Urn $B$ has $3$ white and $12$ black balls. We flip a fair coin. If the outcome is heads, then a ball from urn $A$ is selected, whereas if the outcome is tails, then a ball from urn $B$ is selected.

02

Solution

Urn $A$ $5$ white

$7 B l a c k 鈬� P (w h i t e) = \frac{5}{12}$

Urn $B$ $3$ White

$12 B l a c k 鈬� P (w h i t e) = \frac{3}{15}$

$P (Tail | white) = \frac{\frac{1}{2} 脳 \frac{3}{15}}{\frac{1}{2} 脳 \frac{3}{15} + \frac{1}{2} 脳 \frac{5}{12}}$

$= \frac{3}{30} 脳 \frac{240}{74}$

$= \frac{12}{37}$

03

Final answer

The probability that the coin landed tails is $\frac{12}{37}$ .

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Most popular questions from this chapter

A total of 500 married working couples were polled about their annual salaries, with the following information resulting:

For instance, in 36 of the couples, the wife earned more and the husband earned less than \( 25,000. If one of the couples is randomly chosen, what is

(a) the probability that the husband earns less than \) 25,000 ?

(b) the conditional probability that the wife earns more than \( 25,000 given that the husband earns more than this amount?

(c) the conditional probability that the wife earns more than \) 25,000 given that the husband earns less than this amount?

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?

Assume, as in Example $3 h$ , that percent of twins are of the same sex. Given that a newborn set of twins is of the same sex, what is the conditional probability that the twins are identical?

On rainy days, Joe is late to work with probability $. 3$ ; on nonrainy days, he is late with probability $. 1$ . With probability $. 7$ , it will rain tomorrow.

(a) Find the probability that Joe is early tomorrow.

(b) Given that Joe was early, what is the conditional probability that it rained?

Consider a school community of $m$ families, with $n_{i}$ of them having $i$ children, $i = 1, 鈥�, k, {鈭�}_{i = 1}^{k} n_{i} = m$ Consider the following two methods for choosing a child:

$1$ . Choose one of the $m$ families at random and then randomly choose a child from that family.

$2$ . Choose one of the ${鈭�}_{i = 1}^{k} i n_{i}$ children at random.

Show that method $1$ is more likely than method $2$ to result

in the choice of a firstborn child.

Hint: In solving this problem, you will need to show that

${鈭�}_{i = 1}^{k} i n_{i} {鈭�}_{j = 1}^{k} \frac{n_{j}}{j} 鈮� {鈭�}_{i = 1}^{k} n_{i} {鈭�}_{j = 1}^{k} n_{j}$

To do so, multiply the sums and show that for all pairs $i, j$ , the coefficient of the term $n_{i} n_{j}$ is greater in the expression on the left than in the one on the right.

See all solutions

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