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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?

Any hints please?

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    $\begingroup$ 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 $1/2$. $\endgroup$ – Andreas Caranti Apr 7 '15 at 16:10
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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 $4$.

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  • $\begingroup$ For 1, Is it $0.5^5$? it can't be that simple... $\endgroup$ – shinzou Apr 7 '15 at 16:12
  • $\begingroup$ Why can't it be that simple? It is. $\endgroup$ – Ross Millikan Apr 7 '15 at 22:33

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