Probability: Tree diagrams

​​In a nutshell

Tree diagrams are used to find probabilities of multiple events. They give a visual representation of probabilities.

Drawing tree diagrams

To draw tree diagrams, draw branches associated with each possible outcome, and write down the probability of each outcome next to the associated branch.

Replacement

When writing down probabilities for a repeated trial, be cautious of whether or not there is replacement in the trial.

For example, if a question specifies picking coloured balls out of a box without replacement, it means that there are less balls in the box after each pick, meaning that the denominator of the probabilities will not stay the same.

Finding probabilities

To find probabilities with a tree diagram, multiply the corresponding probabilities along the desired branch.

If the desired event consists of two or more branches, add up the probabilities at the ends of the branches.

Example 1

There are $$7$$ red marbles and $$3$$ green marbles in a box. Alex takes a marble out of the box without replacement, notes down the colour, and takes another marble out of the box.

i) Draw a tree diagram with the probabilities.

ii) What is the probability that Alex picked $$2$$ red marbles?

iii) What is the probability that Alex picked $$1$$ red marble and $$1$$ green marble?

Part i):

Remember that 'without replacement' means that there are only $$9$$ marbles in the box for the second pick.

Part ii):

​Multiply the probabilities along the two red branches:

​$$\begin{aligned}P(\text{2 reds}) &= \dfrac{7}{10}\times \dfrac{6}{9}\\ \\&=\dfrac{42}{90}\\ \\&=\dfrac{7}{15}\end{aligned}$$​​

​$$P(\text{2 reds})=\underline{\dfrac{7}{15}}$$​​

Part iii):

Multiply the probabilities along the red-green and green-red branches, then add them:

​$$\begin{aligned}P(\text{1 red, 1 green}) &= P(\text{(red,green) or (green,red)}) \\&=P(\text{red,green}) + P(\text{green,red})\\&=\dfrac{7}{10}\times\dfrac{3}{9}+\dfrac{3}{10}\times\dfrac{7}{9}\\ \\&=\dfrac{21}{90}+\dfrac{21}{90}\\ \\&=\dfrac{42}{90}\\ \\&=\dfrac{7}{15}\end{aligned}$$​​

​$$P(\text{1 red, 1 green})=\underline{\dfrac{7}{15}}$$ 

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