# Confusing puzzle in probability

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?

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

• I see... thanks Sep 14, 2019 at 12:13
• 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)
– Ned
Nov 13, 2019 at 15:04
• For question 1 no. The answer is $p/7$. For question 2, yes, the answer is $p$ since $P(\text{Floor 7} | \text{In Building and not on 1-6})=1$ Nov 13, 2019 at 15:09
• @DavidPeterson You considered that the second question asks to find the probability $p("Floor~~7")$ while the user Ned didn't understand the same Nov 13, 2019 at 18:32
• @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! Nov 13, 2019 at 18:35