## Probability III – Probability Spaces & Probability Trees

Before you read on, have you read our article on Probability – Multiple Events?

Probability Spaces

When working out the probability of two things occurring, it can often be helpful to draw a probability space. This can help you visualise the problem you are given.

Example 1

You have two fair dice. What is the probability of throwing a total score greater than 8?

There are 6 possibilities for the first dice, and another 6 for the second, so we can draw ourself a 6×6 probabibity square. We now need the total number that add up to greater than 8. So the lowest number for your first throw must be a 3, followed by a 6 on your second throw. Next is a 4 for your first throw, followed by a 5 or 6 on your second throw, etc.

Following this method, we fill in our probability space like such: Now we can read off the total number of ways to get more than 8. There are 10 coloured squares, out of a possible 36.

Hence, the probability of throwing a score greater than 8 with two fair dice is $\frac{10}{36} = 2.78$ to 2dp.

Drawing our probabilities like this can really help in visualising a problem. However probability are limited to two events, such as flipping a coin twice or rolling a dice twice. We’ll not take a look at a way of representing more than two events.

Probability Trees

Another way of representing multiple events is a probability tree.

A probability tree starts with a different branch (hence the name tree) representing each different option. Then for the second set of options, each branch has each option again branching off it.

For example, if it was the score of a fair dice we were interested we were interested in, then there would be initially 6 branches, each representing the numbers 1-6. Each of these 6 branches would then have another 6 new branches coming off them, each representing the number of the second roll. We’ll take a look at an example now to try and make things a bit more understandable.

Example 2

You flip a fair coin two times. What is the probability of getting a different results on each flip, (ie getting either HT or TH)?

Now this is a relatively easy question and I’m sure you could answer it without using a probability tree. But bare with me and we’ll see how useful a probability tree can be!

Firstly we draw our tree, and it should look something like this: Notice that at each stage, the total sum of probabilities is always 1.

From the first set of branches, we can see that the $P(\mathrm{Heads}) = P(\mathrm{Tails}) = \frac{1}{2}$.

It also follows that $P(\mathrm{Head \; and \; Head}) = P(\mathrm{Tail \; and \; Tail}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.

Also note that It also follows that $P(\mathrm{Head \; and \; Tail}) = P(\mathrm{Tail \; and \; Head}) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{2}$.

So we can see that our answer of getting different results is $\frac{1}{2}$.

Now this may seem trivial, using a probability tree when we could just count the results on our hands. But what if we flipped the coin 5 times instead of 2. Or what if it was rolling a dice with six options at each “branch”. The beauty of a probability tree is that it makes all these options easy to visualise, not matter how many times you repeat the event.