How do you calculate the conditional probability using a probability tree?

To calculate conditional probability using a probability tree, multiply along the branches and divide by the total probability.

A probability tree is a visual tool that helps you map out all possible outcomes of a sequence of events. Each branch represents a possible outcome and its probability. To find the conditional probability, you first identify the path that represents the event you're interested in. Multiply the probabilities along this path to get the joint probability of the sequence of events.

For example, let's say you have a probability tree for two events: A and B. Event A can either happen (A1) or not happen (A2), and the same goes for Event B (B1 and B2). The tree will have branches for each combination: A1B1, A1B2, A2B1, and A2B2. To find the conditional probability of B1 given A1, you first find the joint probability of A1 and B1 by multiplying the probabilities along the A1B1 path.

Next, you need the total probability of A1 happening. This is found by adding the probabilities of all branches that start with A1 (i.e., A1B1 and A1B2). Finally, divide the joint probability of A1 and B1 by the total probability of A1. This gives you the conditional probability P(B1|A1).

Using a probability tree makes it easier to visualise and calculate these probabilities, ensuring you account for all possible outcomes and their respective probabilities.