What's the probability distribution for this problem?

Question

In my city there is a bus stop, during 6:30pm to 7pm there is a scheduled bus, the time it arrives the stop is random but it always comes in this time range, what will be the probability distribution when the random variable is the time? I have two contradicted pictures in my mind: it's uniform distribution since the probability that the bus come in every minute is equal; it's some other distribution since the probability that the bus comes is higher and higher in later time of 6:30-7pm if the bus haven't come in previous time range.

Answer 1

I believe your two "contradicting" images are actually not contradicting. They are just answers to two different questions.

Pay close attention to your own words: the probability that the bus comes is higher and higher in later time of 6:30-7pm if the bus haven't come in previous time range. You are talking about conditional distributions here.

Let's suppose that the distribution is (originally) uniform over the 30 minutes, so if you go to the bus stop at 6:30, you think that there is $1/30$ chance that it arrives during the first minute, $1/30$ for the next and so on, up to $1/30$ chance that it will arrive during the last minute. Adding up you have $30 \cdot (1/30) = 1$ probability = certainty that it will arrive before 7:00.

Now if you have waited for 20 minutes and the bus did not come, then you know that this time it will be in the remaining 10-minute range. The conditional distribution is uniform over the remaining range. You have $1/10$ chance that it will arrive during the next minute. Remember: This is because now you know it did not arrive between 6:00 - 6:20. Originally you did not know that.

If you have waited for 29 minutes and the bus has not come, you are certain (probability = 1) it will arrive during the last minute. Originally you had just $1/30$ chance that it would be during the last minute. Two different probabilities because one is unconditional and the other is conditional.

If you have waited for 29 minutes and 59 seconds, and the bus did not come, you are certain it will arrive during the next second. (If your assumptions are valid.)