Lesson

A **tree diagram** is useful in tracking compound events. At each point it branches out to all the possible events that could occur from that point.

It is especially useful if the events have different weightings or are unequal events.

If each outcome is equally likely the probability may be left off, if the outcomes are not equally likely then the probability will be written on the branches.

The sum of the probabilities for the branches from a single node should sum to 1.

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

Here are some examples. Neither 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 occurring:

Notice that the sum of the branches from a single point always adds to 1 (or 100\%). This indicates that all the outcomes are listed.

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.

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

a) \text{P(win, lose)} + \text{P(lose, win)} + \text{P(win, win)} = 21\% + 21\% + 9\% = 51\% or

b) Use the complementary event of losing both games and calculate:

1 - \text{P (lose, lose)} = 1- 49\% = 51\%

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.

One type of experiment that is dependent on previous trials is an experiment without replacement. This means that the object selected (e.g. card, marble, person) is not able to be selected in any other selections.

Three fair coins are tossed.

a

Find the probability of obtaining at least one head.

Worked Solution

b

What is the probability of obtaining TTH in this sequence?

Worked Solution

c

What is the probability of obtaining THH in this sequence?

Worked Solution

The proportion of scholarship recipients at a particular university is \dfrac{7}{10}. The number of students at the university is so large that even if a student is removed, we can say that the proportion of scholarship recipients remains the same. If three students are selected at random:

a

What is the probability that at least one of the students is a scholarship recipient?

Worked Solution

b

What is the probability that at least one of the students is a nonrecipient?

Worked Solution

c

What is the probability there is at least one recipient and one nonrecipient in the selection?

Worked Solution

Idea summary

A **tree diagram** is useful in tracking two-step experiments. It is named because the diagram that results looks like a tree.

For tree diagrams:

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 percentage should add to 100, or the final fractions should add to 1 - this is useful to see if you have calculated everything correctly.

Describe the results of two- and three-step chance experiments, both with and without replacements, assign probabilities to outcomes and determine probabilities of events. Investigate the concept of independence.