Three people get into an empty elevator at the first floor of a building that has 10 floors. Each presses the button for their d

Question

3 months ago

6

Three people get into an empty elevator at the first floor of a building that has 10 floors. Each presses the button for their d

esired floor (unless one of the others has already pressed that button). Assume that they are equally likely to want to go to floors 2 through 10 (independently of each other). What is the probability that the buttons for 3 consecutive floors are pressed?

Mathematics

1 answer:

Zina [12.3K] · Answer 1 · 2026-03-07T10:34:04+03:00

Answer:

The chance of the buttons for three consecutive floors being pressed is $\frac{1}{12}$ or 0.0833

Step-by-step explanation:

Consider the information given.

The users have an equal probability of selecting floors 2 through 10.

Thus, the quantity of buttons is 9.

Each person will press the button for their desired floor (unless someone else has already selected that floor).

This implies that if multiple individuals aim for the same floor, only one will push the button.

Therefore, the total arrangements are calculated as: 9 × 8 × 7 = 504.

Choosing three consecutive floors means they can be:

2,3,4 or 3,4,5 or 4,5,6 or 5,6,7 or 6,7,8 or 7,8,9 or 8,9,10.

With three individuals, they can be arranged in 3! different ways.

As a result, the total number of favorable outcomes in this scenario is 7 × 3! = 42.

$Probability = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

$Probability = \frac{42}{504}=\frac{1}{12}$

Hence, the probability that the buttons for 3 consecutive floors get pressed is $\frac{1}{12}$ or 0.0833