*Posted on December 23, 2011 by Anuj Varma*

A probability problem – will these two friends ever meet?

Two friends decide to meet at a definite place between 9 AM and 10 AM. The first to arrive waits for his friend for 15 minutes and then leaves. If each of them arrives at an arbitrary moment between 9 and 10 AM, find the probability that they will meet.

### First Step – Sample Space

We can treat this problem geometrically. The interval from 9:00 AM to 10:00 AM represents a segment of length 1. Let a unit length segment on the x axis represent one friend’s sample space. Let a unit segment on the y axis represent the other friend’s sample space. The total sample space is then an area of unit length squared. Every point in this unit square represents the arrivals of the two friends.

### Geometric Interpretation of the condition

The events(x,y) during which the friends actually meet are given by the condition

(1)

### Solution

The intersection of the above band with the unit square gives us the required probability (7/16)

### Related Problems

- Two points A and B are thrown at random onto a unit segment of the x axis. Find the probability that the length of the segment will be smaller than the distance between the origin and the nearest point
- Buffon’s needle problem –A plane is ruled with parallel straight lines a distance
*a *apart. A needle of length *l *where *l < a* is thrown on the plane at random. Find the probability that the needle will hit any of the lines.

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