Six buses come to a particular stop each hour, so on average there is a bus every ten minutes.?

Question

If you go to the bus stop at a random time, which is more likely?

a) You will have to wait less than ten minutes
b) You will have to wait about ten minutes
c) You will have to wait longer than ten minutes

Answer 1

c

Answer 2

It depends on the actual bus schedule. If the schedule was evenly distributed, then you'd have to wait an average of 5 minutes (e.g. less than 10 minutes). If you just missed it, you'd have to wait 10 minutes, but otherwise it would be less and the average would be 5.

However, let's say the buses were scheduled as follows:
Bus 1: On the hour 12:00, 1:00, 2:00, etc.
Bus 2: One minute after 12:01, 1:01, 2:01, etc.
Bus 3: Two minutes after 12:02, 1:02, 2:02, etc.
Bus 4: Three minutes after 12:03, 1:03, 2:03, etc.
Bus 5: Four minutes after 12:04, 1:04, 2:04, etc.
Bus 6: Five minutes after 12:05, 1:05, 2:05, etc.

So if you randomly walked into the bus stop, only 8% of the time would you hit that period where the buses were arriving every minute. The other 92% of the time, you'd be waiting for bus #1 to come on the hour. And that wait would be between 0 and 55 minutes (an average of 27.5 minutes).

So here the expected wait time would be 11/12 * 27.5 + 1/12 * 0.5 = 25 1/4 minutes, in other words longer than 10 minutes.

So again it depends.
If the schedule is evenly distributed, then the answer is a) you will wait less than ten minutes.
If the schedule is *unevenly* distributed, it could be any answer listed.

Answer 3

You do not have enough information to answer this question. Here is why:

You only know that six buses arrive every hour. That's the rub. You could schedule the arrival of the six buses any way you like during that hour, and it would still average out to six per hour. As a matter of fact, it MUST everage out to six per hour since all six arrive every hour. Saying it's 'one per ten minutes on average' is the same as saying it's '144 per day on average'.

And that's the problem. All six buses might arrive at the exact same time, or a different one might arrive every ten minutes. We have no way of knowing which is the case, since we have no other information about the arrival of the buses, just our own arrival.

If you like, we can make assumptions about the bus schedules. But they will be just guesses.

We can assume that the buses arrive at a random time in each hour, but that they each arrive once per hour. If any bus has an equal chance of showing up in each minute, then each bus has a 10 in 60 chance of showing up during a ten minute wait, and a 50 in 60 chance of not showing up. The chances that NO bus will show up is 50 in 60 six times over, or about 33.5%. So if the buses arrive at some random time in the hour, then you have a 66.5% chance of catching one in ten minutes. It's statistically likely to wait less than ten minutes, making the answer (a).

We can assume that the buses arrive exactly every ten minutes. If that's the case, then we'll NEVER have to wait MORE than ten minutes (which makes option c seem kind of ridiculous). The break-even point would be at five minutes, which is to say half the time we'll wait more than five minutes, and half the time we'll wait less. That means statistically this is the average length of our wait, so your answer again would by (a).

Of course the worst case scenario is if they all arrive at the same time (a perversity of city planning, I'm sure). The break-even point in this case would be thirty minutes with our own random arrival. More likely than not our wait would be much more than ten minutes, making the correct answer (c).

I leave it to you which conjecture you wish to follow. Perhaps the person asking the questions makes implicit questions such as this all the time, so you know what he MEANS, even if he didn't SAY it. Good luck! ( :

Answer 4

a. You'll wait less than 10 minutes.

A bus comes around every 10 minutes, so unless you show up the moment the last bus pulls away, your wait will likely be less than 10 minutes. Statistically, you have around a 1% chance of waiting 10 minutes or longer.

Answer 5

Based on the information given I think it's mostly guess work. However, I would suggest answer b.

b) You will have to wait about ten minutes.

Answer 6

A
If they average 10 mins apart
you would be waiting 9 mins or less

Answer 7

Probably (a). However, you can drown walking across a lake which has an average depth of 3 feet.

Answer 8

a

Answer 9

a

Answer 10

a)