Suppose that the times until Hector Ivan and JacobSolutionH

Suppose that the times until Hector, Ivan, and Jacob

Solution

H: Time until hector gets his pizza; I: time until Ivan gets his pizza; J: time until Jacob gets his pizza.

The probability that at least one of them did not get their pizza within 20 minutes is equal to 1 minus the probability that all of them got their pizza within 20 minutes.

Since they are independent:

The probability that all of them get their pizza within 20 minutes,

P(H < 20 and I < 20 and P < 20) = P(H < 20)*P(I < 20)*P(J < 20)

Since H, I, and J are exponentially distributed, P(X < a) = 1 - e^(-a/mean)

P(H < 20) = 1 - e^(-20/10) = 1 - e^-2

P(I < 20) = 1 - e^(-20/20) = 1 - e^-1

P(J < 20) = 1 - e^(-20/20) = 1 - e^-1

P(H < 20)*P(I < 20)*P(J < 20) = (1 - e^-2)*(1 - e^-1)*(1 - e^-1) = .3455

So the probability that all of them get their pizzas within 20 minutes is .3455

And the probability that at least one does not get his pizza within 20 minutes is 1 - .3455 = 0.6545

Answer: 0.6545