John needs to catch a train. The train arrives randomly some time between 2:00 and 3:00, waits for 20 minutes, and then leaves. If John also arrives randomly between 2:00 and 3:00, what is the probability that the train will be there when John arrives?
John has a 100% chance of being there if he arrives at 2:20 or earlier. Add 1/3.
At 3:00, he has a 1/3 chance. We can take this and take the average from 2:20 and 3:00 to get 2/3 for this 40 minute period. Multiply this by the portion of time. Add 4/9 to the probability.
Result: He has a 7/9 chance of catching the train.
See Alan's answer No.#2 here. Make sure to adjust for the time difference of 20 minutes instead of 10 minutes in this particular question: p=1 - (1/2(60 - 20)^2/60^2 =(1/2*40^2)/60^2 =1 - 2/9 =7/9: