I would like to introduce probability trees. These help compute more complex probabilities and combine the addition and multiplication rules we covered earlier. Let’s look at a probability tree for two tosses of a coin:

To create the tree, you start with a branch for the first set of possible events for the first trial, in this case heads or tails, then add other branches for all the possibilities for the second trial and so on. You also include the probabilities on each branch segment.

Travelling along a branch depicts a joint probability. For example, what is the the probability of tossing two heads? Travelling along the branches for two heads, you hit two probabilities, each 0.5. As we saw before, this is a joint probability so we multiply these together to get 0.25. So along a branch, you multiply the probabilities.

What if I asked what is the probability of getting two heads or two tails? The only two branch paths that satisfy this requirement are the top one and the bottom one, each with a calculated branch probability of 0.25. These are then added, since this is an OR probability so the addition rule applies:

Adding these probabilities gives the result of 0.50.

Now let’s look at the marble experiment: picking in succession two marbles in a bag with 10 red and 10 blue marbles. Now we can build the probability tree but to know what the probabilities are, we need to know if the first marble is replaced or not. From the last post, you saw that this affects the probabilities of the second pick. I’ll leave it as an exercise for you to build the tree for the “with replacement” case. It will be similar to the coin toss tree.

Without replacement, the tree will look like this:

I’ve kept the probabilities as fractions to make it clearer where they came from. See my last post if needed. Notice that the last column of probabilities add up to 1 as they should since all possible branches have been included.

So what is the probability of picking two blue marbles? This can be read directly from the tree as 0.237. Now let me ask, what has the greater probability: picking two marbles of the same colour or two of different colours? If you add the probabilities of picking two blue or two red, you get 0.474, not quite 50% as you might expect. To get the probability of getting mixed marbles, you can either add the two tree probabilities of 0.263 or subtract the 0.474 probability we just calculated from 1 as this is the only other possibility as the two events are mutually exclusive. Both ways will give the result 0.526. So which possibility would you choose if you were a betting person?

In my next post, I will use probability trees to show the surprising result of the Monte Hall experiment.