12.05 Tree diagrams and multi-stage events

Multistage events

Multistage experiments are those that incorporate two or more simple experiments, for example tossing a coin and rolling a die. Finding probabilities of multistage events is easier if we use a list, table, or tree diagram to show all possible outcomes.

Tables

A table is useful for showing all possible outcomes of two events in the rows and columns. For example, if we tossed $1$1 coin and $1$1 die we can show the outcomes for the coin along the first column and the outcomes for the die across the top row.

Each cell in the table is an outcome of rolling a die and a coin. There are $12$12 possible outcomes in the sample space.

 

Worked example

Two fair dice are thrown and their difference recorded.

a) Use a table to list all the possible outcomes.

We can arrange the outcomes of the first die along the top row and the second die outcomes down the first column then fill in the cells.

 

b) How many outcomes are in the sample space?

The outcomes fill a $6$6 by $6$6 table which means there are $36$36 possible outcomes

 

c) What is the probability of a difference of $1$1?

An outcome of $1$1 occurs $10$10 times out of the total $36$36 outcomes. Therefore $P$P( $1$1 ) = $\frac{1}{36}$136​

Tree diagrams

A tree diagram is named because the diagram that results looks like a tree.

Sometimes calculating probabilities can be complicated, especially if the events have different weightings or are unequal events.

A tree diagram has 4 important components

• Seed
• Branch
• Probability (used when the events have unequal probabilities)
• Outcome

 

Note: Often tree diagrams with the probabilities on the branches are called "Probability Tree Diagrams" while tree diagrams without the probability on the branches are simply called "Tree Diagrams".

Probability tree diagrams are drawn when the outcomes are not equally likely.

 

Trees for single stage events

When a single trial is carried out, we have just one column of branches.

Here are some examples. None of these have probabilities written on the branches because the outcomes are equally likely.

Here are some examples that have probabilities on the branches, because they do not have an equal chance of occuring.

 

Something to notice about tree diagrams.

• When looking at a group of branches, the sum of the group should add to $1$1. This indicates that all the outcomes are listed.

Trees for multistage events

When more than one experiment is carried out, we have two (or more) columns of branches.

Here are some examples. These ones do not have the probabilities written, because the outcomes are equally likely.

Probability tree diagrams

Here is an example that have probabilities on the branches. This probability tree diagram shows the outcomes of playing two games of tennis where the probability of winning is $\frac{3}{10}$310​ and the probability of losing is $\frac{7}{10}$710​.

The probabilities of the events are multiplied along each branch, for example the probability of winning both games is $9%$9% which is found by $\frac{3}{10}\times\frac{3}{10}=\frac{9}{100}$310​×310​=9100​

To find the probability of at least 1 win, I could do either

a) P (win, lose) + P (lose, win) + P (win, win) = $21%+21%+9%=51%$21%+21%+9%=51%

or b) use the complementary event of losing both games and calculate: 1 - P (lose, lose) = $1-49%=51%$1−49%=51%

 

Things to note from using probability trees to calculate probabilities of multiple trials.

• Multiply along the branches to calculate the probability of individual outcomes.
• Add down the list of outcomes to calculate the probability of multiple options.
• The final % should add to 100, or the final fractions should add to 1 - this is useful to see if you have calculated everything correctly.

 

Practice questions

Question 1

Question 2

 

Tree diagrams for dependent events

For multistage events where the next stage is affected by the previous stage, we call these dependent events. We need to take care when drawing the tree diagram accordingly.

Practice question

Question 3

 

Probability tree diagrams of events "without replacement"

In an experiment where the outcome of the first stage is reduced. For example, if we draw an ace card out of a standard deck of $52$52 cards, leave the card out, and draw the second card; then there are only $3$3 ace cards left in the deck of $51$51 cards to draw from.

Practice question

Question 4