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Math Help - Please could someone explain this probability question!

  1. #1
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    Exclamation Help with probability question please!

    Question :
    Suppose a train driver has a 2% chance of crash during every journey. Can we say that he would be mathematically certain of crash after 45 journeys?


    I am trying to apply the combinations formula in this question as a tree diagram would be too time consuming, I am now thinking i am going down the COMPLETELY wrong path! Could anyone please take a few minutes to give me some guidance, thank you!!
    Last edited by Ben91; October 21st 2011 at 05:23 AM.
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  2. #2
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    Re: Help with probability question please!

    If A and B are two events, let A\land B denote the event when both A and B occur, and A\lor B denote the event when either A or B (or both) occur. Then P(A\lor B)=P(A)+P(B)-P(A\land B). So, in general the probability of A\lor B is smaller than the sum of probabilities.

    On the other hand, if A and B are independent, then P(A\land B)=P(A)P(B). So, if crashes during different journeys are independent, then the probability that there will be no crash during 45 journeys is (1-0.02)^{45}.
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  3. #3
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    Re: Help with probability question please!

    Hello, Ben91!

    Suppose a train driver has a 2% chance of crash during every journey.

    Can we say that he would be mathematically certain of crash after 45 journeys?

    No!

    The key phrase is "every journey".

    For each journey: . \begin{Bmatrix}P(\text{crash}) &=& 0.02 \\ P(\text{no crash}) &=& 0.98 \end{Bmatrix}

    We have:
    . . P(\text{No crashes in 342 journeys}) \;=\;(0.98)^{342} \;\approx\;0.001

    Therefore:
    . . P(\text{at least one crash in 342 journeys}) \;=\;1 - 0.001 \;=\;0.999 \;=\;99.9\%

    We can never be 100% certain of a crash.

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  4. #4
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    Re: Help with probability question please!

    As usual, I was completely over thinking the question! Big thank you to both of you for explaining the question and allowing me to understand
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