On time to school

The probability that Andi is late is $\begin{aligned} P(\text{Bus and late}) + P(\text{Walk and late}) &= P(\text{Bus})\times P(\text{late}\mid\text{Bus}) + P(\text{Walk})\times P(\text{late}\mid\text{Walk}) \\ &= \frac{1}{4}\times\frac{2}{3} + \frac{3}{4}\times\frac{1}{3} \\ &= \frac{5}{12} \end{aligned}$

The probability that Andi is late two consecutive days is therefore $\left(\dfrac{5}{12}\right)^2 = \dfrac{25}{144}$ and the probability that he's on time at least one of those days is simply the probability that he's not late both days: $1-\frac{25}{144} = \boxed{\dfrac{119}{144}}$

There are $4$ different ways he can get to school two consecutive days: he takes the bus then walks (which has a $\frac{1}{4} \times \frac{3}{4} = \frac{3}{16}$ chance of happening), he walks and then walks (which has a $\frac{3}{4} \times \frac{3}{4} = \frac{9}{16}$ chance), he takes the bus and then takes the bus $\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$ chance), or he walks then takes the bus $\frac{3}{4} \times \frac{1}{4} = \frac{3}{16}$ chance).

Now, we want to find the probability that he will be on time for at least one day. This is equal to $1 - \text{Chance that he is late both days}$ .

We know we have $4$ different ways he can get to school, and so we can use those probabilities to find out the probability that he'll be late both days.

1) He takes the bus then walks, and is late both days ( $\frac{3}{16} \times \frac{2}{3} \times \frac{1}{3} = \frac{6}{144}$ )

2) He walks then walks, and is late both days ( $\frac{9}{16} \times \frac{1}{3} \times \frac{1}{3} = \frac{9}{144}$ )

3) He takes the bus then takes the bus, and is late both days ( $\frac{1}{16} \times \frac{2}{3} \times \frac{2}{3} = \frac{4}{144}$ )

4) He walks then takes the bus, and is late both days ( $\frac{3}{16} \times \frac{2}{3} \times \frac{1}{3} = \frac{6}{144}$ )

Total probability he is late both days: $\frac{6}{144} + \frac{9}{144} + \frac{4}{144} + \frac{6}{144} = \frac{25}{144}$ .

And so the total probability that he is on time at least one of the days: $1 - \frac{25}{144} = \frac{119}{144}$