AElsaeyed_Discussion6

Five people get on an elevator that stops on five floors. Assuming that each has an equal probability of going to any one floor, find the probability that they all get off on different floors.

Given: - The number of people \(n = 5\). - The number of floors \(n = 5\).

The probability \(P\) that all five people get off on different floors is calculated as:

\[ P = \frac{\text{Number of outcomes that fulfill this requirement}}{\text{Total number of possible outcomes}} \]

Where: - The number of favorable outcomes is the number of ways to arrange \(n\) people on \(n\) different floors, which is a permutation of \(n\) taken \(n\) at a time, calculated as \(n!\).

\[ \text{Number of outcomes that fulfill this requirement} = n! = 5! \]

• The total number of possible outcomes is the number of ways \(n\) people can choose among \(n\) floors, with the possibility of choosing the same floor, which is \(n^n\).

\[ \text{Total number of possible outcomes} = n^n = 5^5 \]

Thus, the probability \(P\) is:

\[ P = \frac{5!}{5^5} \]

Substituting the values:

\[ P = \frac{120}{3125} \]

\[ P = 0.0384 \]

Therefore, the probability that all five people get off on different floors is 0.0384, or 3.84%.