Probability trees are very similar to frequency trees, except probability trees show the probability of an event occurring and frequency trees show the number of times that the event in question occurs.

Let’s have a look at an example of a probability tree.

**Example 1**

A bag contains 3 green balls and 4 red balls. We pick a ball out of the bag twice and replace the ball that we pick out. Draw a probability tree for the outcomes.

The two different outcomes when we pick a ball out of the bag are green and red. This means that the first part of the probability tree will have two branches because there are two different possible outcomes; green and red. We write the probabilities of each of the respective event occurring on each of the branches. There are 3 green balls and 4 red balls, which means that there are 7 balls in total. Therefore, the probability of picking a green ball is ^{3}/_{7} and the probability of picking a red ball is ^{4}/_{7}. The first part of the probability tree is given below.

The probabilities on the branches coming from the same point must add up to 1. We can see that this is the case above because ^{3}/_{7} + ^{4}/_{7} = 1. (this rule is important to remember, and we will need it for an example in the next section).

The ball that we picked out the bag is replaced, which means that there are still two outcomes (green and red) and there are the same number of each of the coloured balls. This means that the probability of these outcomes occurring is exactly the same as it was before; the probability of picking a green on the second pick is ^{3}/_{7} and the probability of picking a red on the second pick is ^{4}/_{7}. The whole probability tree looks like what is shown below.

Probability tree diagrams are really useful as they allow us to find the different outcomes that can occur and also find the probability of these different outcomes occurring. In order to find the different possible outcomes, we follow the branches.

I am going to following the top two lines. When we follow these lines, we see that a green ball was picked on the first pick and a green ball was picked on the second pick (GG).

We can do the same for the other 3 outcomes and all of the outcomes are shown on the probability tree diagram below.

We work out the probability of each of these different outcomes occurring by multiplying together the probabilities on the branches that get you to the outcome that you are finding the probability of. For example, let’s find the probability of picking a red ball and then a green ball (RG). This is the third outcome down. In order to find the probability of picking a red then green, we multiply the probabilities on the branches. The calculation is shown below.

When we multiply fractions, we multiply straight across. We multiply both of the numerators to obtain the numerator in the answer, and we multiply both of the denominators to obtain the denominator in the answer.

We can therefore see that the probability of picking a red ball first and then a green ball second is ^{12}/_{49}.

We can do the same for all 3 of the other outcomes and the working is shown below.

Probability trees are very useful when we are asked to find out the probabilities of events occurring that can occur in a few different ways. For example, let’s find the probability that at least one red ball is chosen.

The four different outcomes when we pick two balls are GG, GR, RG and RR. Three of these outcomes have a red ball in them (GR, RG and RR). Therefore, we can find the probability of at least one red ball being chosen by adding the probability of GR, RG and RR together.

We now need to check whether this fraction can be simplified. There are no common factors between 40 and 49, which means that this fraction is already in its simplest form. Therefore, the answer to this question is ^{40}/_{49}.

Alternatively, we could have found the probability of at least one red ball being chosen by using the following formula:

The only outcome that does not contain at least one red ball is the picking of two green balls (GG) and the probability of two green balls being picked is ^{9}/_{49} (^{3}/_{7} x ^{3}/_{7}).

We can see that this is the same answer as above.