Further probability - Intermediate & Higher tier – WJECTree diagrams and conditional probability

When we have a situation where we are considering several events, it is beneficial to have a way of representing it visually. Tree diagrams are visual representations of the outcomes of events.

Example one

Robin has 2 bags. Bag A has 7 balls of which 3 are red and 4 are blue and bag B has 8 balls of which 5 are red and 3 are blue.

Robin is going to take a ball from each bag, he wants to know what all the possible outcomes are so he draws a tree diagram:

Notice how for bag A there are two possibilities, either a red ball or a blue ball can be selected and Robin has put the probability of each choice onto the diagram.

For bag B there are also two possibilities - these are written twice, once for each outcome of the first choice.

We can choose to take several different routes through the diagram depending on which outcomes we are interested in.

What is the probability Robin selects 1 blue ball and 1 red ball?

Solution

There are two ways to select 1 blue and 1 red. P(R+B) or P(B+R) both satisfy the condition. We can use the OR rule to calculate the probability:

Often we use tree diagrams to model conditional probability. This is where there is more than one outcome and they are not independent – in other words the first outcome affects the probability of the second.

Example two

Rachel is going to select a bead from a bag, then select a second bead from the same bag. She is not going to replace the bead in the bag after taking 1 out. In Rachel’s bag there are 4 red beads and 5 green beads.

Notice how the probabilities are different from the first and second choices. If a red bead is chosen first, then there is now 1 less red bead to choose, so the chance of selecting a blue bead is increased and the chance of selecting a second red bead decreases. If a green bead is selected, the chance of getting a green bead the second time decreases, and the chance of getting a red bead increases.