In this activity, we are going to explore the use of tree diagrams for finding the probability of a series of events.

Given the following **tree diagram**, how could we find the probability that a bus is late on both days?

We want the probability that the bus is late on Monday **and **on Tuesday, i.e. the probability of the green path below:

Anytime we are finding the probability of something **'and'** something happening or the probability on the **same path** on the tree diagram, we **multiply **the probabilities:

P(late and late) = 0.3 x 0.3 = **0.09**

Let's say we now wanted to find the probability that the bus is late on exactly one day.

We have two possible paths to achieve such a scenario:

The orange path gives us:

P(on time and late) = 0.7 x 0.3 = **0.21**

The green path gives us:

P(late and on time) = 0.3 x 0.7 =** 0.21**

We want the probability that the bus is late on exactly one day, i.e either on time on Monday and late on Tuesday (orange path) **or **late on Monday and on time on Tuesday (green path).

Anytime we are finding the probability of something **'or'** something happening or the probability of **different paths** on the tree diagram, we **add **the probabilities:

**P(late on one day) = P(on time and late) + P(late and on time) = 0.21 + 0.21 = 0.42**

In this example, the orange and green paths' probabilities were the same - be careful because this will not always happen, we just had a scenario where it did!

Ready to put this all into practice?