### Tree Diagrams

Tree diagrams are a helpful tool for calcuating probabilities when there are several independent events involved. They get their name because these types of diagrams resemble the shape of a tree. The branches of a tree split off from one another, which then in turn have smaller branches. Just like a tree, tree diagrams branch out and can become quite intricate.

If we toss a coin, assuming that the coin is fair, then heads and tails are equally likely to appear. As these are the only two possible outcomes, each has probability of 1/2 or 50%. What happens if we toss two coins? What are the possible outcomes and probabilities? We'll see how to use a tree diagram to answer these questions.

Before we begin we should note that what happens to each coin has no bearing on the outcome of the other. We say that these events are independent of one another. As a result of this, it doesn't matter if we toss two coins at once, or toss one coin, and then the other. In the tree diagam, we will consider both coin tosses separately.

### First Toss

Here we illustrate the first coin toss. Heads is abbreviated as "H" in the diagram and tails as "T". Both of theses outcomes have probability of 50%. This is depicted in the diagram by the two lines that branch out. It is important to write the probabilities on the branches of the diagram as we go. We'll see why in a little bit.

### Second Toss

Now we see the results of the second coin toss. If heads came up on the first throw, then what are the possible outcomes for the second throw? Either heads or tails could show up on the second coin. In a similar way if tails came up first, then either heads or tails could appear on the second throw.

We represent all of this information by drawing the branches of second coin toss off of *both* branches from the first toss. Probabilities are again assigned to each edge.

### Calculating Probabilities

Now we read our diagram from left to write and do two things:

- Follow each path and write down the outcomes.
- Follow each path and multiply the probabilities.

The reason why we multiply the probabilities is that we have independent events. We use the multiplication rule to perform this calculation.

Along the top path, we encounter heads and then heads again, or HH. We also multiply:

50% x 50% = (.50)x(.50)=.25=25%.

This means that the probability of tossing two heads is 25%.

We could then use the diagram to answer any question about probabilities involving two coins. As an example, what is the probability that we get a head and a tail? Since we were not given an order, either HT or TH are possible outcomes, with a total probability of 25%+25%=50%.