One of the classic problems of Mathemagistics, or Mathematical Magic, is the Bus Problem. It goes like this:

Question:

Every day, Fred gets the solitary 8 am bus to work. There is no other bus that will get him to his destination.

10 per cent of the time the bus is early and leaves before he arrives at 8 am.

10 per cent of the time the bus is late and leaves after 8.10 am.

The rest of the time the bus departs between 8 am and 8.10 am.

One morning Fred arrives at the bus stop at 8 am, sees no bus, and waits for 10 minutes without the bus arriving.

Now, what is the probability that Fred’s bus will still arrive?

Fred’s bus could yet arrive or he might have missed it. So there are two possibilities. So is it correct to assume that in the absence of further evidence the chance of each must be equal, so the probability at 8.10am that his bus will still arrive is 50 per cent?

But if that is the answer at 8.10am, was it also the correct answer at 8 am?

Or was 50 per cent the correct answer at 8am but not at 8.10am?

Or is it the wrong answer at both times, but was correct at 8.05am?

The solution is posted below.

Solution

When  Fred arrives at 8am, there is a 10 per cent chance that his bus will have already left. After Fred has waited for 10 minutes, he can eliminate the 80 per cent chance of the bus arriving in the period between 8 am and 8.10 am. So only two possibilities remain.

Either the bus has arrived ahead of schedule or it will arrive more than ten minutes late.

Both outcomes are unusual, but since the two outcomes are mutually exclusive and equally likely (10 per cent chance of each), and there are no other possibilities, we should update the probability that the bus will still arrive from 10 per cent (the likelihood, or prior probability, when Fred woke up) to 50 per cent, as there is (once the 80 per cent probability is eliminated) an equal probability (out of the remaining 20%) that the bus will still turn up and that he has missed it. So there is a 1 in 2 chance that he will still catch his bus if he has the patience to wait further, and a 1 in 2 chance that he will wait in vain. The follow-up question is how long he should wait. That’s for another day.