Chapter 13.2, Problem 12PPS

To determine

To find: the probability that a zip code is randomly generated from a given number and the number is $39372$ .

Answer to Problem 12PPS

The probability is $\frac{1}{3360}$ .

Explanation of Solution

Given:

Zip code randomly generated from the digits are:

  $3,7,3,9,5,7,2\text{ and }3$ .

Concept used:

The probability formula is used to compute the probability of an event to occur as:

  $P\left(A\right)=\frac{\text{Number of favorable outcome}}{\text{Total Number of favorable outcomes}}$ .

According to the permutation formula:

  $P{}_r=\frac{n!}{\left(n-r\right)!}.$

Where $P$ is the permutations.

  $n$ is the total number of objects in the set.

  $r$ is the total number of objects in the set.

Calculation:

There are $8$ digits in which $3$ appears three times and $7$ appears twice.

The number of distinguishable permutations is:

  $\frac{8!}{2!3!}=3360.$

The total number of possible outcomes is $3360$ and there is only one favorable outcome which is $39372$ .

Therefore, the probability is $\frac{1}{3360}$ .

Hence, the probability is $\frac{1}{3360}$ .