Probability of bus arriving at given time

probability

Am i calculating this correctly. The metro stops by every 8 minutes and the bus departs every 10 minutes. If i don't follow the schedules but arrive at the stop randomly. With what probability.

A) I have to wait for the metro more than 5 minutes.

$3/8 = 37.5%$

B) Both metro and bus leave after waiting less than 2 minutes.

$2/8 * 2/8 = 6.25%$

Best Answer

Let's assume that you will only arrive "on the minute" (e.g. at 10:05, never at 10:05 and 15 seconds).

A) When you arrive, there are $8$ possibilities: you arrive $0$ minutes before the metro leaves (i.e. when it is leaving; in this case I'll assume you can get on the metro), you arrive $1$ minute before the metro leaves, you arrive $2$ minutes before the metro leaves, ..., you arrive $7$ minutes before the metro leaves. You will have to wait more than $5$ minutes only if you arrive $6$ minutes before the metro leaves or $7$ minutes before it leaves. Since each possibility is equally likely, and $2$ of the $8$ possibilities require waiting more than $5$ minutes, the probability is $2/8$.

(Aside: your answer of $3/5$ would be the probability of waiting at least $5$ minutes)

B) This one is more complicated. Let's assume the first bus and metro arrive at the same time (say time $0$). Then a metro arrives at times $0, 8, 16, 24, 32, 40 = 0$. A bus arrives at times $0, 10, 20, 30, 40 = 0$. In both cases I treat $40$ as "wrapping around" to $0$ because that's when the full cycle of metro-bus arrivals repeats.

If both the metro and the bus leave after less than $2$ minutes of waiting, then they must leave within less than $2$ minutes of one another (for example, if they left with a $3$ minute gap between them you'd have to wait for at least $3$ minutes). The only time this happens is at minute $0=40$ in the cycle. If you want to wait for less than $2$ minutes, you'd need to arrive at either minute $39$ or minute $40$. So there's a $2/40$ probability.

If you're interested in the probability of waiting at most $2$ minutes (i.e. you're okay if you have to wait $2$ minutes, rather than less than $2$ minutes). Now you can achieve this by arriving at minute $8$, minute $30$, minute $38$, minute $39$, or minute $40$. Since each of these $5$ arrivals is equally likely, you have a probability of $5/40$.