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Math Help - Probability 2

  1. #1
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    Probability 2

    A town has three bus routes A, B and C. Route A has twice as many buses as each of B and C. Over a perido of time it has been found that, along a certain stretch of road, where the three routes converge, the buses on these routes run more than five minutes late 1/2, 1/5 and 1/10 of the time respectively.

    1) A bus is going down this stretch of road. What is the probability it is more than 5 minutes late?

    2) Comment on the size of the answer to 1) with the respect to the given probabilities 1/2, 1/5 and 1/10.

    3) An inspector, standing on this particular stretch of road, sees a bus that is more than five minutes late. Find the probability that it is a route B bus.

    Help please
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Natasha
    A town has three bus routes A, B and C. Route A has twice as many buses as each of B and C. Over a perido of time it has been found that, along a certain stretch of road, where the three routes converge, the buses on these routes run more than five minutes late 1/2, 1/5 and 1/10 of the time respectively.

    1) A bus is going down this stretch of road. What is the probability it is more than 5 minutes late?
    Part 1)

    There are twice as many rout A buses as either route B or C, so half of the
    buses are route A. So the proportion P(A) of buses which are route A is 0.5, and
    the proportion P(B) that are route B is 0.25, and the proportion P(C) route C
    is again 0.25. So the probabilty that a random bus is late is:

    <br />
P(late)=P(A)P(late|A) + P(B)P(late|B)+P(C)P(late|C)<br />

    But we are told that the proportion P(late|A) of route A buses which are late is 1/2=0.5,
    and that P(late|B)=1/5=0.2, and P(late|C)=1/10=0.1. So:

    <br />
P(late)=0.5 \times 0.5 + 0.25 \times 0.2+0.25 \times 0.1=0.325<br />

    RonL
    Last edited by CaptainBlack; February 18th 2006 at 10:45 AM.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Natasha
    A town has three bus routes A, B and C. Route A has twice as many buses as each of B and C. Over a perido of time it has been found that, along a certain stretch of road, where the three routes converge, the buses on these routes run more than five minutes late 1/2, 1/5 and 1/10 of the time respectively.

    1) A bus is going down this stretch of road. What is the probability it is more than 5 minutes late?

    2) Comment on the size of the answer to 1) with the respect to the given probabilities 1/2, 1/5 and 1/10.
    Part 2)

    Answer to 1) is 0.325, which is less than the proportion of route A buses
    which are late because the lateness of route A is diluted by the better
    time keepers of routs B and C, but only slightly because half of all buses
    are the poor time keepers of route A.

    RonL
    Last edited by CaptainBlack; February 18th 2006 at 10:44 AM.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Natasha
    A town has three bus routes A, B and C. Route A has twice as many buses as each of B and C. Over a perido of time it has been found that, along a certain stretch of road, where the three routes converge, the buses on these routes run more than five minutes late 1/2, 1/5 and 1/10 of the time respectively.

    1) A bus is going down this stretch of road. What is the probability it is more than 5 minutes late?

    2) Comment on the size of the answer to 1) with the respect to the given probabilities 1/2, 1/5 and 1/10.

    3) An inspector, standing on this particular stretch of road, sees a bus that is more than five minutes late. Find the probability that it is a route B bus.
    Part (3).

    This is simply an aplication of Bayes theorem:

    <br />
P(B \wedge late)=P(B)P(late|B)=P(B|late)P(late)<br />

    So we may write:

    <br />
P(B)P(late|B)=0.25 \times 0.2
    <br />
 P(B|late)P(late)=P(B|late)\times 0.325<br />

    So:

    <br />
P(B|late)=\frac{0.25 \times 0.2}{0.325}=0.154<br />

    RonL
    Last edited by CaptainBlack; February 18th 2006 at 11:48 AM.
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  5. #5
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    Thanks Ron!
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