Tree Diagrams

Introduction: Combined Probability

Combined probability means there are two or more events occurring.

For two events, these can be split into two SEPARATE events, and then a rule can be applied to work out the overall probability:

1. The AND rule
2. The OR rule

Events may be independent – the outcome of one does not affect the outcome of another; or dependent – the outcome of one does affect the outcome of another.

The AND Rule

Where event one and event two are independent events, the AND rule means both event one AND event two occur. Let’s call event one A and event two B.

The probability of Event A and Event B both happening is the same as the probability of Event A happening multiplied by the probability of Event B happening, or:

\text{P}(A \text{ and } B) = \text{P}(A)\times\text{P}(B)

 

Example:

If the probability of rain is 0.7 and the probability of thunder is 0.2, what is the probability of rain AND thunder?

Call probability of rain, \text{P}(R) = 0.7, and thunder, \text{P}(T)=0.2

Use the AND equation:

0.7\times0.2=0.14

So, the probability of rain AND thunder is 0.14

The OR Rule

The OR rule means either Event A OR Event B occur, where both Events are mutually exclusive (this means, the events cannot happen at the same time) 

The probability of Event A OR Event B happening is the same as the probability of Event A happening added to the probability of Event B happening, or:

\text{P}(A \text{ and } B) = \text{P}(A)+\text{P}(B)

 

Example:

A biased dice is thrown with the probabilities given below,

Calculate the probability of a 4 or a 6 being rolled.

Use the OR equation:

0.31+0.16=0.47

So, the probability of rolling a 4 OR a 6 is 0.47

Tree Diagrams – Independent Events

Firstly, let’s look at how to construct and understand tree diagrams where the events are independent.

For example, a bag contains 3 red counters and 7 blue counters.

Lauren picks 2 counters at random. After she picks the first, she puts it back in the bag and then randomly takes the next. This means these events are independent, as the probabilities will be the same the second time she chooses a counter.

Then, we can follow this red line to the second selection, where we meet the options red and blue again. If Lauren gets a blue counter on her second selection, this probability is \dfrac{7}{10}.

Therefore, using the AND rule:

\text{P}(A)\times\text{P}(B)=\dfrac{3}{10}\times\dfrac{7}{10}=0.21

 

Using the OR rule:

Example: find the probability Lauren chooses two counters of the same colour.

There are two ways Lauren could do this, she could get two red counters, or two blue counters.

To find the probability of two red counters, we need to firstly use the AND rule (as she is getting a red counter AND a red counter):

\dfrac{3}{10}\times\dfrac{3}{10}=0.09

Same for two blue:

\dfrac{7}{10}\times\dfrac{7}{10}=0.49

Now, she either gets two reds OR two blues, so let’s use the OR rule:

\text{P}(A)+\text{P}(B)=0.09+0.49=0.58

Tree Diagrams – Dependent Events

To draw tree diagrams for dependent events, you need to be more careful when putting the probabilities onto the diagram.

Let’s use the same example as earlier, but let’s say that Lauren does not replace the counters after the first selection.

Remember, the bag contains 3 red counters and 7 blue counters.

On the first selection, with all the counters in the bag, the probabilities will remain,

\text{P}(R)=\dfrac{3}{10}\\

\text{P}(B)=\dfrac{7}{10}\\

If Lauren picks a red counter on her first selection, there will only be 2 red counters left in the bag for her second selection, out of 9 possible counters, so the probabilities would be:

\text{P}(R)=\dfrac{2}{9}\\

\text{P}(B)=\dfrac{7}{9}\\

If Lauren picks a blue counter on her first selection, there will only be 6 bluecounters left in the bag for her second selection, out of 9 possible counters, so the probabilities would be:

\text{P}(R)=\dfrac{3}{9}\\

\text{P}(B)=\dfrac{6}{9}\\

And the tree diagram would be:

The AND rule and the OR rule for dependent events

Earlier, you learnt that the AND rule only work for independent events. However, tree diagrams are slightly different. 

When plotting a dependent tree diagram, you take into account dependence when drawing probabilities onto the tree, and this means you can use the AND rule for any tree diagram.

Hence, you can use the AND rule for both independent and dependent tree diagrams.

For the OR rule, dependent events on a tree diagram are always mutually exclusive,

Hence, you can also use the OR rule for both independent and dependent tree diagrams.