# Probability of an event that occur first of a joint uniform distribution

A man and a woman agree to meet at a certain location about 12:30 P.M. If the man arrives at a time uniformly distributed between 12:15 and 12:45, and if the woman independently arrives at a time uniformly distributed between 12:00 and 1 P.M.

What is the probability that the man arrives ﬁrst?

X: The arrival time of the man u~(15,45)

and

Y: The arrival time of the woman uni~(0,60)

Can anyone explain to me why I have to calculate the $P(X<Y)$ ?

what I want to know is why the event $X<Y$ corresponds to the event that the man arrives ﬁrst.

• The event $X<Y$ corresponds to the event when the man arrives ﬁrst Commented Jul 22, 2015 at 21:46
• @ConradoCosta: What would you answer if the question is: Can anyone explain why I have to calculate the probability that the man arrives first?
– zoli
Commented Jul 22, 2015 at 21:53
• That is a hard question with an easy answer that would go like this the question asks the probability that the man arrives first, so if you want to solve the question, you must calculate it. But I might be missing the point. What would you answer? Commented Jul 22, 2015 at 21:58
• I can't tell you why you have to calculate $P(X<Y)$, but if you want to know the answer to the problem, it's $1/2$ by symmetry about 12:30. Commented Jul 22, 2015 at 22:10
• @ConradoCosta: I am just pulling your leg. joriki already gave the answer: "I can't tell you why you have to calculate P(X<Y)."
– zoli
Commented Jul 22, 2015 at 22:35

• can you explain why the event $X<Y$ corresponds to the event that the man arrives ﬁrst plz? thank you Commented Jul 23, 2015 at 0:05
• @HtangUvong Sorry, I misread “why I have to calculate” as “how I have to calculate” in your question. Let's see: \begin{align*} X=&\,\text{the time at which the man arrives},\\ Y=&\,\text{the time at which the woman arrives}. \end{align*} Now, the event $X<Y$ happens if and only if the time at which the man arrives occurs before the time at which the woman arrives. In plain English, this means that the man arrives earlier than the woman. Commented Jul 23, 2015 at 3:15