Airport shuttle and Jimmy

Question

The airport shuttle arrives at the hotel at a random time between 7.30 am and 7.45 am. Jimmy waits at the hotel a random time between 7.30 am and 7.45 am (independently of the shuttle) and he will wait for (at most) $5$ min before leaving. Find the probability that Jimmy will catch the bus.

So far the way I've thought about this problem is:

$X \sim U[0, 15]$ where $X$ is the probability that the shuttle arrives at a specific minute.

$Y \sim U[0, 15]$ where Y is the probability that Jimmy arrives at a specific minute.

Find $P(Y\le X\le Y+15).$

Does this seem fine? I'm not sure entirely sure.

@SomebodyOnEarth It seems that you have posted the same question yesterday, deleted it, and posted it again. If you have done this, it is not good practice, and this wastes people's time and effort. — Toby Mak, Dec 04 '17 at 23:19
That wasn't me. This is the first time I've posted this question. — SomebodyOnEarth, Dec 04 '17 at 23:21
Is the way I'm thinking about it correct? (It's in the question description) — SomebodyOnEarth, Dec 05 '17 at 00:00
Your setup is fine. You can think of the two arrival times as a point in a square $15$ minutes on a side. Now draw the region where Jimmy catches the bus and compare the area to the area of the square. — Ross Millikan, Dec 05 '17 at 05:25
at a first sight I would say $\frac23 \frac13 + \frac13 \frac16=\frac{5}{18}$ — user, Dec 05 '17 at 08:08
Related: https://math.stackexchange.com/questions/103015/chance-of-meeting-in-a-bar — Henry, Dec 05 '17 at 11:48

score 1 · Accepted Answer · answered Dec 05 '17 at 23:44

1

I managed to solve it by plotting a graph and then calculating the needed area.

To find the area I solved the following integral:

$\int_{0}^{15} \int_{0}^{x} \frac{1}{225}dy \space dx - \int_{5}^{15} \int_{0}^{x-5} \frac{1}{225}dy \space dx $

Which gives an answer of $\frac{5}{18}$

answered Dec 05 '17 at 23:44

SomebodyOnEarth

305

If you are ok, you can set as solved. Thanks! – user Dec 06 '17 at 13:24

score 0 · Answer 2 · answered Dec 05 '17 at 14:36

In general,

indicating with:

$T_J$: maximum time interval Jimmy waits at the hotel (i.e. 5 min in th example)

$T_T$: total time interval (i.e. 15 min in th example)

and with:

$a=\frac{T_J}{T_T}$ the times interval ratio,

the probability that Jimmy will catch the bus should be:

$$P(a)=a-\frac{a^2}{2}$$

In this case:

$$P(\frac13)=\frac13-\frac{1}{18}=\frac{5}{18}$$

Airport shuttle and Jimmy

2 Answers2