# Probability of an elevator rising to a certain floor at most and exactly

In a $$10$$ story building, $$5$$ people enter an elevator on ground level and press the floor buttons $$(1-10)$$ in random and independently.

1. What is the probability that the elevator will rise at most to floor $$5$$

2. What is the probability the elevator will rise exactly to floor $$5$$ and not further up.

1. I think the calculation should be to find the probability for one person getting off at floor 1 and the rest in the other floors below 6, then the same with two people at floor 1 etc and the same for each of the five floors, this is insane to calculate even with a calculator.

So maybe using a complement will work: the elevator rose at least to floor 6 which is enough for one person to choose this floor and the rest went to floors $$\le 5$$ so we have: $$1-(\frac 5 {6})^4$$. (4 people, everyone else going below the 6th).

2. This is like at least one person went to the 5th floor and everyone else went to floors $$\le 5$$. The complement would be: none went to the 5th floor and at least one went to the 6th floor. So $$1-(\frac 4 5)^4$$.

I feel like I'm making a lot of mistakes, why does it have to be only one person to the 6th floor and not another floor or more people?

• For the first one, note that each time a person presses a button, the probability that it does hit one of the buttons $\{ 1, 2, 3, 4, 5 \}$ is $1/2$. Commented Apr 7, 2015 at 16:10

Hint:

For 1, you just need all five buttons to be in the range $1$ to $5$. What is the chance for each button? Then multiply them.

For 2, you need all five buttons to be in the range $1$ to $5$, and to have at least one be $5$. Take your result from part $1$ and subtract the chance that all buttons are in the range $1$ to $4$.

• For 1, Is it $0.5^5$? it can't be that simple... Commented Apr 7, 2015 at 16:12
• Why can't it be that simple? It is. Commented Apr 7, 2015 at 22:33