Chapter 8.6, Problem 57E

(a).

To determine

To find:

The probability when randomly guess the position of each digit.

Answer to Problem 57E

The required probability is $0.008333$ .

Explanation of Solution

Given:

On a game show given five digits to arrange in the proper order to form the price of a car.

Concept Used:

Apply the formula of the probability:

  $\begin{array}{c}P\left(A\right)=\frac{NumberofFavourableOutcome}{TotalNumberofFavourableOutcome}\\ =\frac{n\left(A\right)}{n\left(S\right)}\end{array}$

Calculation:

There are $5$ digits so, the arrangements of these $5$ digits is $5!=120$ ways.

Thus, $n\left(S\right)=120$ .

Only one of the them is correct if guess the position of each digit then $n\left(A\right)=1$ .

Therefore, the probability of winning if guess the positions of each digit is:

  $\begin{array}{c}P\left(A\right)=\frac{1}{120}\\ =0.008333\end{array}$

Conclusion:

Hence, the required probability is $0.008333$ .

(b).

To determine

To find:

The probability when know the first digit and randomly guess the others.

Answer to Problem 57E

The required probability is $0.04166$ .

Explanation of Solution

Given:

On a game show given five digits to arrange in the proper order to form the price of a car.

Concept Used:

Apply the formula of the probability:

  $\begin{array}{c}P\left(A\right)=\frac{NumberofFavourableOutcome}{TotalNumberofFavourableOutcome}\\ =\frac{n\left(A\right)}{n\left(S\right)}\end{array}$

Calculation:

After knowing the first digit there are remaining $4$ digits which can be arranged in $4!=24$ ways.

Thus, $n\left(S\right)=24$ .

Only one of the them is correct then $n\left(A\right)=1$ .

Therefore, the probability of winning if know the first digit:

  $\begin{array}{c}P\left(A\right)=\frac{1}{24}\\ =0.04166\end{array}$

Conclusion:

Hence, the required probability is $0.04166$ .