# Probability that the inpatient guy arrived first

So the question asks:

Tom and Jerry set up a meeting at a restaurant. Each one of them, independently of the other, arrives at some random time between 9:00 pm and 10:00 pm (that is, the arrival time is uniformly distributed between 9 and 10). Jerry is quite patient and waits for 10 minutes and then leaves, while Tom waits for 5 minutes and then leaves. What is the probability that they meet? Given that they met, what is the probability that Tom arrived first?

So so far I got:

So the area between the blue lines is the possibility that Tom meets Jerry:

So P(Tom and Jerry meet) = $\frac{60^2 - 0.5 * 55^2- 0.5 * 50^2}{60^2} = 0.232638889$

And for the probability that Tom arrived first given they would meet, I think since the total area of possibility that they meet is

$60^2 - 0.5 * 55^2- 0.5 * 50^2 = 837.5$

So the probability that Tom arrived first would be : $\frac{0.5 * 837.5 * \frac{5}{10+5}}{837.5} = 0.166666666$

But I really not sure about this answer. So does the first part look alright, and how can I get the probability that Tom arrived first?

• Can you explain your numerator in the second part? The first part looks good. Commented Mar 13, 2016 at 16:27
• The first part looks good. The second part should be close to $\frac 1 3$.
– A.S.
Commented Mar 13, 2016 at 16:27

For the $2_{nd}$ part, compute the favorable area above the main diagonal to the total favorable area
For the second part, $\dfrac{0.5(60^2 - 55^2)}{0.5(60^2-55^2+60^2-50^2)} =\dfrac{23}{67}$
Your answer for the probability they meet is correct. However, I am not sure about your formula for the probability Tom arrived first. It should be $\frac{\text{P(Tom arrives first, then they meet)}}{\text{P(They meet)}}$. You have already solved the denominator (the area between the two blue lines). Now, just solve for the numerator and divide. (Draw a diagonal of the square from the bottom left to the top right vertices. The area between the diagonal and the top blue line would be the probability Tom showed up first.) In other words, you want $\frac{\text{Area between purple and red lines}}{\text{Area between purple and blue lines}}$ in this image:
• I guess you mean $\frac{\text{Area between purple and red lines}}{\text{Area between purple and blue lines}}$ Commented Mar 14, 2016 at 23:31