uniformly distributed random variables

Question

Ram and Shyam wanted to meet at a park about 12.30 P.M.. If Ram arrives at a time uniformly distributed between 12.15 P.M. to 12.45 P.M. and if Shyam independently arrives at a time uniformly distributed between 12.00 P.M. and 1.00 P.M., then find the probability that the first to arrive waits no longer than 5 minutes

Answer 1

Suppose first to arrive has to wait for no longer than 5 minutes then the condition would of wait time would become 0 to 5 minute. Ram has to come from 12.15 to 12.45 Shyam has to come from 12.10 to 12.50 to maintain wait time no longer than 5 minutes As per the rule of uniform distribution

      P = 1/(b-a). f(X) dx

     So sample space would be time from 12.10 P.M to 12.50 PM

     Therefore, a = 10, b = 5
                   P = 1/(50-10). f(X)

                  P =1/(40) . (0-5)

                  P = 5/40 = 1/8

                  Probability (P) = 1/8     

Ans : So probability of first to arrive wait no longer than 5 minute would be 1/8.