Chapter 13.3, Problem 41E

(a)

To determine

To find: The probability that Carl can solve all $10$ problems on the test.

Answer to Problem 41E

The probability that Carl can solve all $10$ problems on the test is $\frac{21}{1292}$ .

Explanation of Solution

Given:

There 20 study problems.

Calculation:

The order is important here, there are $C\left(20,10\right)$ ways to choose the 10 questions. There are $C\left(15,10\right)$ ways to choose the 10 questions that can answer and there are $C\left(5,0\right)$ ways to choose the questions that she cannot answer.

The probability that Carl can solve all 10 problems is as follows.

  $\begin{array}{c}{P}_1=\frac{C\left(15,10\right)\cdot C\left(5,0\right)}{C\left(20,10\right)}\\ =\frac{3003\cdot 1}{184756}\\ =\frac{21}{1292}\end{array}$

Therefore, the probability that Carl can solve all $10$ problems on the test is $\frac{21}{1292}$ .

(b)

To determine

To find: The odds that Carl will know to solve 8 of the problems.

Answer to Problem 41E

The odds that Carl will know to solve 8 of the problems is $\frac{225}{421}$ .

Explanation of Solution

Given:

There 20 study problems.

Calculation:

Let $A$ be the event.

Carl will know how to solve 8 of the problems determine the probability of success of event $A$ .

Order is not important in this case.

There are $C\left(15,8\right)$ ways to choose the 8 problems, and $C\left(5,2\right)$ ways to choose the 2 remaining problems.

Find the probability of remaining problem.

  $\begin{array}{c}P\left(s\right)=\frac{C\left(15,10\right)\cdot C\left(5,2\right)}{C\left(20,10\right)}\\ =\frac{6435\cdot 10}{184756}\\ =\frac{64350}{184756}\\ =\frac{225}{646}\end{array}$

Determine the probability of failure of event $A$ .

  $\begin{array}{c}P\left(f\right)=1-P\left(s\right)\\ =1-\frac{225}{646}\\ =\frac{421}{646}\end{array}$

Find the odds as follows.

  $\begin{array}{c}\text{odds}=\frac{P\left(s\right)}{P\left(f\right)}\\ =\frac{\frac{225}{646}}{\frac{421}{646}}\\ =\frac{225}{421}\end{array}$

Therefore, the odds that Carl will know to solve 8 of the problems is $\frac{225}{421}$ .