Question

In Exercises 5–36, express all probabilities as fractions.

Safety with Numbers The author owns a safe in which he stores all of his great ideas for the next edition of this book. The safe “combination” consists of four numbers between 0 and 99, and the safe is designed so that numbers can be repeated. If another author breaks in and tries to steal these ideas, what is the probability that he or she will get the correct combination on the first attempt? Assume that the numbers are randomly selected. Given the number of possibilities, does it seem feasible to try opening the safe by making random guesses for the combination?

Short answer

The probability that the correct combination is guessed on the first attempt is equal to $\frac{1}{100000000}$.

As the total number of different possibilities is extremely large, it is not feasible to guess the correct combination of numbers on the first try.

Step-by-step solution

Given information

A safe has a code consisting of four numbers selected from 0 to 99 with repetition.

Counting principle

The counting principle suggests that the number of possibilities for an event to occur iscounted at each step and then multiplied to get the total number of possibilities for that event to happen.

Calculation

Let A be the event of getting the correct combination of numbers.

The number of digits from 0 to 99 is equal to 100.

The total number of ways in which four numbers can be selected from 0 to 99 with repetition is shown below:

$$\begin{gathered} \text{Total}\text{number}\text{of}\text{ways}=\text{100}\times \text{100}\times \text{100}\times \text{100} \\ =100000000 \end{gathered}$$

The number of ways of arranging the four numbers to get the correct combination is equal to 1.

The probability of guessing the correct combination of numbers is shown below:

$P\left(A\right)=\frac{1}{100000000}$

Therefore, the probability of guessing the correct combination of numbers is equal to $\frac{1}{100000000}$.

Since theprobability value is extremely low(approximately equal to 0), it can be said it is not feasible to guess the correct combination of numbers on the first try.