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10.03 Tree diagrams


Tree diagrams

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.

Tree diagram showing first light encountered at two sets of traffic lights

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

The important components of the tree diagram are:

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

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$1.


Trees for single stage events

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.

The outcomes for tossing a coin once Outcomes from rolling a standard die

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

Showing outcomes of whether the Hawk or the Tigers win the match

Outcomes of selecting a face card or number card from deck of cards

Outcomes of certain person playing a game of chess


Notice that the sum of the branches from a single point always adds to $1$1 (or $100%$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.

Tree diagram for tossing a coin three times Tree diagram for whether a child gets home during the day or night during the next three days

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 $0.3\times0.3=0.09$0.3×0.3=0.09

To find the probability of at least $1$1 win, we 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%$149%=51%


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


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.

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. 

For example, the probability of drawing a red card from a standard pack of $52$52 cards is $\frac{26}{52}=\frac{1}{2}$2652=12. If we do draw a red card and choose to select a second card without replacement there are only $25$25 red cards left, but there are still $26$26 black cards. And there are only $51$51 cards left in the entire deck. The probability of selecting a second red card is $\frac{25}{51}$2551. This can be seen in the top branches of the tree diagram above.



Practice questions

Question 1

Three fair coins are tossed.

  1. Find the probability of obtaining at least one head.

  2. What is the probability of obtaining TTH in this sequence?

  3. What is the probability of obtaining THH in this sequence?

Question 2

The proportion of scholarship recipients at a particular university is $0.7$0.7. 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:

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

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

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



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.

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