Exercise 4.50. Suppose an alarm clock has been set to ring after T hours, where T Exp(l/3). Suppose further that your friend has been staring at the clock for exactly 7 hours and can confirm that it has not yet rung. At this point, your friend wants to know when the clock will finally ring. Calculate her conditional probability that she needs to wait at least 3 more hours, given that she has already waited 7 hours. More generally, calculate the conditional probability that she needs to wait at least x more hours, given that she has already waited 7 hours.

a) P(T>10 | T>7) = 0.3678

b) P(T>7+x | T>7) =

Step-by-step explanation:

a)Conditional Probability that she needs to wait at least 3 more hours, given that she has already waited 7 hours

Three more hours means x = 7+3 = 10

P(T>10 | T>7) = P ( (T>10) ∩ (T>7))  / P (T>7)

P ( (T>10) ∩ (T>7)) = P (T>10)

P (T>10) = 1 -  P (T≤10)

P (T≤10) =

P (T≤10) = 0.9643

P (T>10) = 1 -  P (T≤10) = 1 - 0.9643

P (T>10) = 0.03567

 P (T>7) = 1 -  P (T≤7)

P (T≤7) =

P (T≤7) = 0.903

P (T>7) = 1 -  P (T≤7) = 1 - 0.903

P (T>7) = 0.097

P(T>10 | T>7) =  0.03567/0.097

P(T>10 | T>7) = 0.3678

b) The conditional probability that she needs to wait at least x more hours, given that she has already waited 7 hours

P(T>7+x | T>7) = P ( (T>7+x) ∩ (T>7))  / P (T>7)

P ( (T>7+x) ∩ (T>7)) = P (T>7+x)

P (T>7+x) = 1 -  P (T≤7+x)

P (T≤7+x) =

P (T>7+x) = 1 -  P (T≤(7+x)) = 1 -

P (T>7+x) =

P (T>7+x) =

P (T>7+x) =

P (T>7) = 0.097

P(T>7+x | T>7) =

P(T>7+x | T>7) =

c

step-by-step explanation: