Unique Floor Exits

Solution & Explanation

In this problem, we are tasked with calculating the probability that each of four passengers in an elevator gets off on a different floor of a building with four floors. To solve this, we need to consider both the total number of possible outcomes and the number of favorable outcomes where each passenger exits on a distinct floor.

Step 1: Determine the Total Number of Outcomes

Each of the four passengers has the option to get off on any of the four floors. Therefore, the total number of ways the passengers can choose floors is calculated as:

$4 \times 4 \times 4 \times 4 = 4^4 = 256$

This is the total number of possible combinations of floor selections for the four passengers.

Step 2: Determine the Number of Favorable Outcomes

A favorable outcome occurs when each passenger exits on a different floor. To find this, we consider the following:

• The first passenger can choose any of the 4 floors.
• The second passenger can choose any of the remaining 3 floors (since one floor is already taken).
• The third passenger can choose any of the remaining 2 floors.
• The fourth passenger has only 1 floor left to choose.

Thus, the number of favorable outcomes is calculated as:

$4 \times 3 \times 2 \times 1 = 4! = 24$

Step 3: Calculate the Probability

The probability that each passenger gets off on a different floor is the ratio of the number of favorable outcomes to the total number of possible outcomes:

$P(\text{each passenger exits on a different floor}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{24}{256}$

Simplifying the fraction gives:

$\frac{24}{256} = \frac{3}{32}$

Conclusion

Therefore, the probability that each of the four passengers exits the elevator on a different floor is $\frac{3}{32}$. This result is derived by recognizing the factorial arrangement of passengers exiting on distinct floors and dividing by the total possible combinations of exits.