Two friends agree to meet at a park with the following conditions. Each will reach the park between 4:00 pm and 5:00 pm and will see if the other has already arrived. If not, they will wait for 10 minutes or the end of the hour whichever is earlier and leave. What is the probability that the two will NOT meet?
GATE 1998 · Engineering Mathematics · Uniform Distribution · medium
Set up the sample space: (X, Y) is uniform over [0,60] x [0,60]. Total area = 3600.
Identify non-meeting region: Two triangular regions: (1) X - Y > 10: triangle with vertices (10,0), (60,0), (60,50). (2) Y - X > 10: triangle with vertices (0,10), (0,60), (50,60).
Compute area of non-meeting region: Each triangle: area = (1/2) * 50 * 50 = 1250. Total non-meeting area = 2 * 1250 = 2500.
Compute probability they do NOT meet: P(NOT meet) = 2500 / 3600 = 25/36 = 0.6944...