Question

Waiting on the platform, a commuter hears an announcement that the train is running five minutes late. He assumes the arrival time may be modeled by the random variable T, such that

f(T = t) = {3/5 (5/t)^4 , t ≥ 5
0, otherwise
If given the train arrived in less than 15 minutes, what is the probability it arrived in less than 10 minutes?
А. 62%
B. 73%
C. 88%
D. 91%
E. 96%

Answer 1

D. 91%

Explanation:

Conditional Probability

We use the conditional probability formula to solve this question. It is

`P(B|A) = (P(A \cap B))/(P(A))`

In which

P(B|A) is the probability of event B happening, given that A happened.

`P(A \cap B)` is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

Event A: Less than 15 minutes.

Event B: Less than 10 minutes.

We are given the following probability distribution:

`f(T = t) = (3)/(5)((5)/(t))^4, t \geq 5`

Simplifying:

`f(T = t) = (3*5^4)/(5t^4) = (375)/(t^4)`

Probability of arriving in less than 15 minutes:

Integral of the distribution from 5 to 15. So

`P(A) = \int_(5)^(15) = (375)/(t^4)`

Integral of
`(1)/(t^4) = t^(-4)` is
`(t^(-3))/(-3) = -(1)/(3t^3)`

Then

`\int (375)/(t^4) dt = -(125)/(t^3)`

Applying the limits, by the Fundamental Theorem of Calculus:

At
`t = 15`,
`f(15) = -(125)/(15^3) = -(1)/(27)`

At
`t = 5`,
`f(5) = -(125)/(5^3) = -1`

Then

`P(A) = -(1)/(27) + 1 = -(1)/(27) + (27)/(27) = (26)/(27)`

Probability of arriving in less than 15 minutes and less than 10 minutes.

The intersection of these events is less than 10 minutes, so:

`P(B) = \int_(5)^(10) = (375)/(t^4)`

We already have the integral, so just apply the limits:

At
`t = 10`,
`f(10) = -(125)/(10^3) = -(1)/(8)`

At
`t = 5`,
`f(5) = -(125)/(5^3) = -1`

Then

`P(A \cap B) = -(1)/(8) + 1 = -(1)/(8) + (8)/(8) = (7)/(8)`

If given the train arrived in less than 15 minutes, what is the probability it arrived in less than 10 minutes?

`P(B|A) = (P(A \cap B))/(P(A)) = ((7)/(8))/((26)/(27)) = 0.9087`

Thus 90.87%, approximately 91%, and the correct answer is given by option D.