What's the probability of truck A arrives before truck B

Question

Truck A arrives at a random time between 9am and 11am, and truck B arrives at a random time between 10am and 12pm (noon). what are the odds that truck A arrives before truck B??

I searched this question and followed the hint provided by 'drhab':

Let X be the arrival time of truck a and let Y be the arrival time of truck b. If X,Y are two random variables on the same probability space then:

    Pr(X<Y)=∫Pr(X<Y∣Y=y)dFY(y)=∫Pr(X<y∣Y=y)dFY(y)

If moreover X and Y are independent then:

     Pr(X<y∣Y=y)=Pr(X<y)

so that:

     Pr(X<Y) = ∫Pr(X<y)dFY(y)

I got final result is 0.75, just want to confirm this outcome. Thanks!

Answer 1

Let's divide the problem up into several possible outcomes:

• $X_1$: A arrives between 9am and 10am (B at any time).
• $X_2$: A arrives between 10am and 11am and B arrives between 11am and 12am.
• $X_3$: Both A and B arrive between 10am and 11am.

These three are disjoint and together they give all possible results. So let now $R$ be the result that $A$ arrives before $B$. Then we have $$P(R) = P(R | X_1) + P(R | X_2) + P(R | X_3).$$

Are you ok with it so far? Now let's look at the $X_i$: The chance of $X_1$ should better be $1/2$, if we are assuming the time to be completely random. The chance of $X_2$ happening is $1/4$, as we have $1/2$ chance that $A$ arrives in this time slot and the same for $B$. As in both $X_1$ and $X_2$, we always have $A$ arriving before $B$, we can say $$P(R) = 1/2 + 1/4 + P(R | X_3).$$

As the first two terms already sum up to $0.75$, for your result to be true you would need $P(R | X_3) = 0$. But it is well possible that $A$ arrives before $B$ and they still both arrive between 10am and 11am...

Answer 2

The pair of arrival times is a random point in the square $Q:=[9,11]\times[10,12]$ with probability measure ${\rm d}P={1\over4}{\rm d}({\rm area})$. The line $x=y$ cuts off a small triangle at the lower right of $Q$. If the random point is above this line truck $A$ is first, and if the random point is below this line truck $B$ is first. It follows that the probability in question is ${7\over8}$.