Probability of two persons to meet Karan and Arjun planned to meet in a park between 3 pm and 5 pm. It was decided that whoever comes earlier will wait for the other person for 40 minutes or till 5 pm which ever comes first. What is the probability that both will meet?
 A: At $3$, start a timer at $0$. We need to assume something about the arrival times of our two heroes. Let their arrival times (in terms of the timer) be $X$ and $Y$. We will assume that $X$ and $Y$ are independent and uniformly distributed in the interval $[0,2]$. We want the probability that $|X-Y|\le  \frac{2}{3}$. 
Draw the square with corners $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$. Draw the lines $y-x=\frac{2}{3}$ and $x-y=\frac{2}{3}$. We want the probability that $(X,Y)$ ends up in the part of the square between the two lines. Call this region $A$.
Since the pair $(X,Y)$ is uniformly distributed in the square, and the square has area $4$, our required probability is the area of $A$ divided by $4$.
Look at the picture you have drawn. The region $A$ differs from the square by two triangles. It should be easy to find their combined area. The area of $A$ will then be $4$ minus the combined area of the two triangles. 
A: Let K be the time that Karan arrives and A be the time that Arjun arrives.
Left F = min(K,A) be the time the first one arrives and S = max(K,A) be the time the second arrives.
They will meet if F is later than 4:20pm or S - F is less than 40 minutes.  The probability of this event depends on the distribution of K and A.
