Chapter 8.6, Problem 61E

(a)

To determine

To find; the probability of winning when the first digit and randomly guess the others.

Answer to Problem 61E

The probability of winning if guess the position of given digit is $\frac{1}{120}$ .

Explanation of Solution

Given information:

On a game show, given five digits to arrange in the proper order to form the price of a car, If are correct, win a car.

Calculation:

Let us try to find that guess the position of given digit for all five digits position of first digit can be guess in 5 ways.

Position of second digit can be guessed in 4 ways, since the first digit has been guessed and four digits are remaining

Thus, position of third digit can be guessed in 3 ways

Position of fourth digit can be guessed in 2 ways

And position of fifth digit can be guessed in 1 way

The above can be shown as follows

5

4

3

2

1

Therefore all 5 digits can be arranged in following ways

  $\begin{array}{l}=5\times 4\times 3\times 2\times 1\\ =120\text{ways}\end{array}$

Since only one way is correct, the probability of winning if guess the position of given digit is $\frac{1}{120}$ .

(b)

To determine

To find; the probability of winning when randomly guess the position of given digit.

Answer to Problem 61E

The probability of winning if first digit is known and guess the position of rest of the digits is $\frac{1}{24}$ .

Explanation of Solution

Given information:

On a game show, given five digits to arrange in the proper order to form the price of a car, If are correct, win a car.

Calculation:

If the first digit is known, then let us try to find the probability of winning, if the rest of the positions of digits have to be guessed

First digit is known. So,

Position of second digit can be guessed in 4 ways

Thus, position of third digit can be guessed in 3 ways

Position of fourth digit can be guessed in 2 ways

And position of fifth digit can be guessed in 1 way

The above can be shown as follows

0

4

3

2

1

Therefore all 4 digits can be arranged in following ways

  $\begin{array}{l}=4\times 3\times 2\times 1\\ =24\text{ways}\end{array}$

The probability of winning if first digit is known and guess the position of rest of the digits is $\frac{1}{24}$ .