Hello, I'm stuck on a problem and I wasn't getting any answer in the pre-university statistic topic so 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).

I thought of something but it doesn't work:

$\displaystyle C_{n}=\{\text{arrives with car on day n}\}$

$\displaystyle A=\{\text{arrives on time}\}$

then if $\displaystyle P$ is our function of probability:

$\displaystyle 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}{2} P(C_{n-1}) + \frac{1}{4}(1-P(C_{n-1})) = \frac{1}{4} (P(C_{n-1})+1) $

then recursively you get

$\displaystyle \frac{P(C_{1})}{4^n} + \sum^{n}_{k=1} \frac{1}{4^k}= \frac{P(C_{1})}{4^n} + \frac{1- \frac{1}{4^{n-1}}}{ \frac{3}{4}} $

but if n---->inf then $\displaystyle P(C_n)$ is greater than 1... find the big mistake....

thanks a lot in advance.