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I have this question which is confusing me. I searched a lot to find an explanation but without success, though I am sure that it is a known puzzle and there is someone who asked it before!

We know that, with probability $p$, the person we are looking for lives in a certain $7$-story building. If this person lives in the building, then he is equally likely to live on any of the $7$ floors.

  • What is the probability that he lives on the $7$th floor of this building?

  • We explore without success the first $6$ floors. What is the probability now that he lives on the $7$th level?

How to solve this question? Do we need to assume that the probability is equally-likely? Do we take into consideration that the person lives outside the building?

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You know

I. $P(\text{In building})=p$

II. $P(\text{Floor 7} | \text{In building})=\dfrac{1}{7}$

For question 1, you are looking for:

$P(\text{Floor 7}) = P(\text{Floor 7} | \text{In building})P(\text{In building})+P(\text{Floor 7} | \overline{\text{In building}})P(\overline{\text{In building}})$

For question $2$ you are finding the same thing, but given the information "the person is not on floors 1 through 6", you know that $P(\text{Floor 7} | \text{In building}) = 1$

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  • $\begingroup$ I see... thanks $\endgroup$
    – user8003788
    Sep 14, 2019 at 12:13
  • $\begingroup$ but hasn't the probability that (s)he lives in the building also changed from the given information? Why not P(Lives on 7 | not lives on 1-6) = P[( L on 7) and (not L on 1-6)] / P (not L 1-6) = P (lives on 7) / P (not live 1-6) = ... = p / (7-6p) $\endgroup$
    – Ned
    Nov 13, 2019 at 15:04
  • $\begingroup$ For question 1 no. The answer is $p/7$. For question 2, yes, the answer is $p$ since $P(\text{Floor 7} | \text{In Building and not on 1-6})=1$ $\endgroup$
    – David P
    Nov 13, 2019 at 15:09
  • $\begingroup$ @DavidPeterson You considered that the second question asks to find the probability $p("Floor~~7")$ while the user Ned didn't understand the same $\endgroup$
    – user8003788
    Nov 13, 2019 at 18:32
  • $\begingroup$ @Ned Why you think the second question asks to find the probability = P(Lives on 7 | not lives on 1-6) why not it asks to find $P(Lives~ on ~7 )$ ? which intuition led you to this? really I am lost now! $\endgroup$
    – user8003788
    Nov 13, 2019 at 18:35

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