5 P) Probability Trees – Part 2
Example 1
A bag contains orange and blue balls. A ball is picked out of the bag twice and replaced after each pick. There is a partially completed probability tree below.
a) Complete the probability tree
b) Find the probability of picking balls of the same colour
Part a
Part a in the question asks us to fill in the probability tree. We are only given one of the probabilities on the probability tree and this is the probability of picking an orange ball on the first pick (probability is 2/5). From this information, we are able to work out the probability of picking a blue ball on the first pick. We are able to do this because we know that the probabilities on the branches from the same point must add up to 1. Therefore, the probability of picking an orange ball on the first pick and picking a blue ball on the first pick must add up to 1. Therefore, we can find the probability of picking a blue ball on the first pick by doing the following calculation.
Therefore, the probability of picking a blue ball on the first pick is 3/5.
We are told in the question that the ball is picked out of the bag and then replaced and this means that the probabilities of picking each of the colours is constant; the probability of picking an orange ball will always be 2/5 and the probability of picking a blue ball will always be 3/5. We can place these probabilities onto the tree.
Part b
Part b asks us to find the probability of a certain event taking place. Before we start answering any questions like this, I think that it is a good to list the outcomes at the end of the branches. We find the outcomes by following the branches on the probability trees. The outcomes are shown on the probability tree below.
We are asked to find the probability of picking two balls of the same colour. From the outcomes column we can see that there are two different outcomes where the balls are the same colour; the balls can either be both orange (first outcome) or both blue (bottom outcome). In order to find the probability of these outcome occurring, we multiply the probabilities on the branches to get to that outcomes.
The final step is to add the probabilities of each of these events happening together. The calculation is shown below.
Therefore, the probability of picking two balls of the same colour is 13/25.
Example 2
Suppose that I have a cake tin that contains 10 cakes. There are 2 chocolate cakes and the rest of the cakes are lemon cakes. I pick a cake out of the tin at random twice and eat them (meaning that the cakes are not replaced). Find the probability of picking one cake of each flavour.
The first step in answering this question is to draw out the probability tree. There are two different outcomes for the first pick; you can either pick a chocolate cake or a lemon cake. There are 2 chocolate cakes and 10 cakes in total, which means that there are 8 lemon cakes. This means that the probability of picking a chocolate cake is 2/10 and the probability of picking a lemon cake is 8/10. Therefore, the first part of the probability tree is like what is given below.
We are now going to deal with the top section of the probability tree, which is where a chocolate cake has been picked on the first pick. When a chocolate cake has been picked, the number of chocolate cakes left in the tin and the number of total cakes left in the tin will decrease by 1; there will now be 1 chocolate cake in the tin and 9 cakes in total. No lemon cakes have been picked and this means that the number of lemon cakes will stay the same; there will be still be 8 lemon cakes. For the second pick from the tin when a chocolate cake has been picked, there are still two different choices of cake that can be picked; chocolate cake or lemon cake. Therefore, we can draw two branches for these outcomes. The probability of picking a chocolate cake after picking a chocolate cake first is 1/9 and the probability of picking a lemon cake when a chocolate cake has been picked first is 8/9. These outcomes and probabilities are shown on the tree below.
We now need to forget about everything that we have just worked out and imagine that we picked a lemon cake first. Before the first pick, there were 2 chocolate cakes, 8 lemon cakes and 10 cakes in total. After a lemon cake has been picked, the number of lemon cakes and the total number of cakes will decrease by 1. The number of chocolate cakes will remain the same because no chocolate cakes were picked. This means that after a lemon cake has been picked on the first pick, there will be 2 chocolate cakes, 7 lemon cakes and 9 cakes in total. We can now work the probabilities out. When a lemon cakes has been picked first, the probability of picking a chocolate cake on the second pick is 2/9 and the probability of picking a lemon cake on the second pick is 7/9. The probability tree with probabilities is shown below.
The next step (which I would always recommend doing) is to write down all of the outcomes on the side of the probability trees. We find the outcomes by following the branches.
- Chocolate cake on the first pick and a lemon cake on the second pick (CL); this is the second outcome down
- Lemon cake on the first pick and a chocolate cake on the second pick (LC); this is the third outcome down
We now need to find the probability of these two outcomes occurring, which we do by multiplying the probabilities on the branches together.
The final step is to add the probabilities of the two different outcomes together to find the probability of picking one cake of each flavour.
As we are giving the probability in fraction form, we need to make sure that our fraction is in its simplest form. We simplify fraction by dividing the numerator and the denominator by the highest common factor between the numerator and the denominator. The highest common factor between 32 and 90 is 2. Therefore, we divide the numerator and denominator by 2.
The probability of picking one cake of each flavour is 16/45.