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## Waiting on the platform, a commuter hears an announcement that the train is running five minutes late. He assumes the arrival time may be mo

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%

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Mathematics
5 months
2021-08-21T06:37:29+00:00
2021-08-21T06:37:29+00:00 1 Answers
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## Answers ( )

Answer:D. 91%

Step-by-step explanation:Conditional ProbabilityWe use the conditional probability formula to solve this question. It is

In which

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

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:Simplifying:

Probability of arriving in less than 15 minutes:Integral of the distribution from 5 to 15. So

Integral of is

Then

Applying the limits, by the

Fundamental Theorem of Calculus:At ,

At ,

Then

Probability of arriving in less than 15 minutes and less than 10 minutes.The intersection of these events is less than 10 minutes, so:

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

At ,

At ,

Then

If given the train arrived in less than 15 minutes, what is the probability it arrived in less than 10 minutes?Thus 90.87%, approximately 91%, and the correct answer is given by

option D.