# Math Help - Conditional probability

1. ## 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).

2. ## 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?