The Meeting

Suppose two people arrange a meeting be 1pm and 2pm. Either can arrive first, at any time between 1pm and 2pm. If one arrives and the other is not there, they will wait up to 30 minutes, with each waiting time being equally likely. If the other one arrives while the first is there they will meet and if the first person leaves they will not meet.<br />
What is the probability the two will meet?<br />
If the first person arrives before 1.30pm he will be there for 30 minutes. If he arrives after 1.30pm he will only be there until 2pm. On average he will be there for 15 minutes - 1/4 of the time.<br />
On average therefore he will be there for 45 out of the 60 minutes between 1pm and 2pm. The probability the two will meet is 45/60=3/4.