Tricky tree diagram probability problem.

Question

$A$ always goes to work by a bus or a taxi. If he goes to work by a bus one day, the probability he goes to work by a taxi the next day is $0.4$. If he goes to work by a taxi one day, the probability he goes to work by a bus the next day is $0.7$.

Given that $A$ takes a bus to work on Monday, find the probability that he takes a taxi to work on Wednesday.

Answer 1

Hint:

The probability tree for the given problem is illustrated below (I leave the filling of the probabilities as an exercise for the OP):

Consider the following two cases:

1. $A$ takes the bus on Monday $\rightarrow$ $A$ takes the bus on Tuesday $\rightarrow$ $A$ takes the taxi on Wednesday
2. $A$ takes the bus on Monday $\rightarrow$ $A$ takes the taxi on Tuesday $\rightarrow$ $A$ takes the taxi on Wednesday

The probabilities of interest are then $\mathbb P\left(B\cap B\cap T\right)$ and $\mathbb P\left(B\cap T\cap T\right)$.