# Probability Question.

Every morning, I roll a die to decide how to travel to work. If I roll a $1$ or $2$, I take the train, if I roll a $3$, I catch a bus and otherwise I cycle. The probability that I am late for work is $1/10$ if I travel by train, $1/5$ if I travel by bus and $1/20$ if I cycle.
I work a 5-day week.

I have worked out the probability i'm late to work is 11/120

(i) If I am late for work, what is the probablity that I travelled by train.

would this simple be $Pr(Travelled by Train | Late)$ hence $4/11$

(i) Calculate the probablity that I am on time every day during a week.

would this be $1-(11/120 * 5)$

(iii) I work for 46 weeks per year. Let Y denote the number of weeks in a year for which I am on time every day of the week. Find the mean and variance of Y .

The probability that you are on time to work on a particular day is $p=1-{11\over120}$. The probability that you are on time for an entire work week is $p^5$ (with the usual independence assumptions), not what you have. For (iii), use the answer to (ii) and think of the binomial distribution (with a "success" being on time for a particular week in the year). –  David Mitra Dec 10 '12 at 17:11
@AndréNicolas for the mean i got mean = $E(Y)$ = np = $46(109/120)^5$ .... is this correct? –  jill Dec 10 '12 at 17:51