Bus A arrives at a random time between 2pm and 4pm, and Bus B arrives at a random time between 3pm and 5pm. What are the odds that Bus A arrives before Bus B?
My understanding is that since Bus B cannot possibly arrive between 2 and 3, we can only talk about the time between 3 and 4 pm, when there is an equal probability for both buses arriving. But in this case, the probability of Bus A arriving before B is 50%. No? What am I missing here? Or I should look at the entire timeline, 2 pm – 5 pm? But then in this case, it is still 50%. Where is my thinking wrong?
Best Answer
Guide:
1) Draw rectangle $2\le x\le 4,$ $3\le y\le 5$.
2) The area of the rectangle is $4$, so pdf is $1/4$.
3) Draw line $y=x$.
4) Find area of the rectangle above the line, which is $7/2$.
5) Finally, the required probability is $7/2\cdot 1/4=7/8$.
Here is the graph:
$\hspace{2cm}$
Bus $A$ arriving before bus $B$: $$A(2.5,4.5),B(3.5,4.5),C(2.5,3.5),D(3.4,3.7).$$ Bus $A$ arriving after bus $B$: $$E(3.7,3.3).$$