Elevator Passenger Probability

Solution & Explanation

To solve the problem of determining the probability that all four individuals will choose different floors to disembark from the elevator, we need to consider both the total number of possible outcomes and the number of favorable outcomes where no two individuals choose the same floor.

Total Number of Possible Outcomes

Each individual can choose from any of the five upper floors to exit. Since there are four individuals, the total number of possible outcomes, where each person makes an independent choice, is calculated as follows:

$5^4 = 625$

This represents the total number of ways the four individuals can choose from the five floors.

Favorable Outcomes (No Two Individuals Choose the Same Floor)

To find the number of ways in which all four individuals choose different floors, consider the following:

• First Individual: Has 5 choices (any of the five floors).
• Second Individual: Must choose a different floor, so has 4 choices remaining.
• Third Individual: Has 3 choices remaining after the first two have chosen different floors.
• Fourth Individual: Has 2 choices remaining after the first three have chosen different floors.

Thus, the number of favorable outcomes where all four individuals choose different floors is calculated as:

$5 \times 4 \times 3 \times 2 = 120$

Probability Calculation

The probability that all four individuals will choose different floors is the ratio of the number of favorable outcomes to the total number of possible outcomes:

$\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} = \frac{120}{625}$

Simplifying the fraction gives:

$\frac{120}{625} = \frac{24}{125} \approx 0.192$

Thus, the probability that all four people will choose different floors to disembark is approximately 0.192, or 19.2%.

Explanation

1. Independence: Each person makes an independent decision regarding which floor to exit.
2. Combination Without Repetition: The problem requires that no two individuals exit on the same floor, which involves calculating combinations without repetition.
3. Sequential Choices: The calculation reflects a sequential choice process where each subsequent individual has fewer choices due to the restriction of not repeating floors already chosen by others.

This problem illustrates a typical probability scenario involving permutations and combinations where order and uniqueness are key factors in determining outcomes.