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Math Help - Conditional probability

  1. #1
    Junior Member
    Joined
    Sep 2009
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    Conditional probability

    Hello, I'm stuck on a problem, I'd be very grateful if someone could help:

    To go to work employs take their car or the bus. If they take their car they have 1/2 chances to be late, if they take the bus only 1/4 chances to be late. If they are on time one day they will take the same mean of transportation the next day, if they are late they switch. If p is the probability that an employ goes to work on day one with his car:

    a) what is the probability that he'll go to work with his car on day n?

    I started by writing the probability with the conditional probability that he went to work on day n-1... but that's the best i can come up with I don't know what else I can do.

    b) what is the probability that he will be late on day n.

    c)what is the limit when n---> inf for a) and b).

    Thanks a lot in advance!!
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  2. #2
    Junior Member
    Joined
    Sep 2009
    Posts
    62

    Re: Conditional probability

    Ok I might have found something.

    C_{n}=\{\text{arrives with car on day n}\}
    A=\{\text{arrives on time}\}

    then if P is our function of probability:
    P(C_n)=P(A|C_{n-1})+P(A^{c}|C_{n-1}^c))=\frac{1}{2} P(C_{n-1})+\frac{1}{4}P(C_{n-1}^c)=\\ \frac{1}{4} (P(C_{n-1})+1)

    so

    P(C_n)= \frac{P}{4} \sum_{k=1}^{n} \frac{1}{4^k}

    am I right?
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