Probability that two persons will meet

Problem Statement: Two persons are supposed to meet a spot in a time interval of one hour. The person who comes first will wait for a maximum of 15 minutes for the second person. What is the probability that they will meet?

Solution: The question can be solved in many ways. Let’s discuss one of them:

It is assumed that the first man arrives the spot during the first 45 minutes i.e first 3/4 hour. The probability that he will arrive during the first 45 minutes is
 \frac{45}{60} = \frac{3}{4} .



Then he will wait for 15 minutes and probability that the second person ill come in next 15 minutes is
 \frac{15}{60} = \frac{1}{4}.

Hence if the first person comes during first 15 minutes, probability that they meet each other is
 \frac{3}{4} *\frac{1}{4} =\frac{3}{16}.

If the first person does not come during first 45 minutes, he will come at next 15 minutes . Probability that he comes during the last 15 minutes is
 \frac{15}{60} = \frac{1}{4} .

If he comes during last 15 minutes, he waits for \frac{1}{8} hours on average.



(This can be shown by the following logic).

Suppose he arrives at the 45th minute. Then he will wait for the next 15 minutes which has a probability of \frac{1}{8}. Again if he arrives at 60th minute he will not wait anymore.

So In case, the first person comes during the last 15 minutes, the probability that the next person comes is \frac{\frac{1}{4}+0}{2}=\frac{1}{8}.

So, in this case probability that they will meet each other is  \frac{1}{4} *\frac{1}{8} =\frac{1}{32} .

So, the total probability that they will meet is \frac{3}{16} +\frac{1}{32}=\frac{7}{32} .

Note that any person can come before anyone. But we have considered only one case.
So, the total probability that they will meet each other is 2*\frac{7}{32}=\frac{7}{16}.



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Improved By : zippa1993