Question

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.

Answer 1

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

b) P(T>7+x | T>7) =
`e^{(-x)/(3) }`

Explanation:

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

`T = e^(1/3)`

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) =
`1 - e^{(-1)/(3) *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) =
`1 - e^{(-1)/(3) *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) =
`1 - e^{(-1)/(3) *(7+x)}`

P (T>7+x) = 1 - P (T≤(7+x)) = 1 -
`(1 - e^{(-1)/(3) *(7+x)})`

P (T>7+x) =
`e^{(-1)/(3) *(7+x)}`

P (T>7+x) =
`e^{(-7)/(3) } * e^{(-x)/(3) }`

P (T>7+x) =
`0.097e^{(-x)/(3) }`

P (T>7) = 0.097

P(T>7+x | T>7) =
`\frac{0.097e^{(-x)/(3) }}{0.097}`

P(T>7+x | T>7) =
`e^{(-x)/(3) }`

Answer 2

P(she needs to wait for 3 more hours | She has already waited for 7 hours) = e⁻¹

P(she needs to wait for x more hours | She has already waited for 7 hours)

= e^(-x/3)

Explanation:

P(she needs to wait for 3 more hours | She has already waited for 7 hours)

= P(she needs to wait for 3 more hours, She has already waited for 7 hours) / P(She has already waited for 7 hours)

= P(she needs to wait for 10 hours) / P(She has already waited for 7 hours)

= (1/3)exp(-10/3)/ (1/3)exp(-7/3) = exp(-1) = e⁻¹

P(she needs to wait for x more hours | She has already waited for 7 hours)

= P(she needs to wait for x more hours, She has already waited for 7 hours) / P(She has already waited for 7 hours)

= P(she needs to wait for 7+x hours) / P(She has already waited for 7 hours)

= (1/3)exp(-(7+x)/3)/ (1/3)exp(-7/3) = exp(-x/3) = e^(-x/3)