So far we have looked into 2 different types of visual representations for data relating to probability, the Venn Diagram and a Two Way Table. We introduce a third type of representation now, the Tree Diagram.
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
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 they all have equal chance of happening.
|The outcomes for tossing a coin once||Outcomes from having a baby||Outcomes from rolling a standard die|
Here are some examples that have probabilities on the branches, because they have specific probabilities of happening.
Things to notice about tree diagrams.
When more than one trial is carried out, we have two (or more) columns of branches.
Here are some examples. These ones without probabilities, because they have equal chance of happening.
tree diagram showing outcomes and sample space for rolling a dice twice
Here is an example that have probabilities on the branches, because they have specific probabilities of happening. The probabilities of the events are multiplied through each branch.
Outcomes of playing two games of tennis. Since I have a 3% chance of winning, the chance that I win 2 games in a row is 9%.
If I wanted to know what the chance is that I win at least 1, I could do either
a) add 9+21+21 together and get 51%,
or b) use the complementary event of losing both games and calculate 1-49 = 51%
Things to note from using probability trees to calculate probabilities of multiple trials.
P(A)+P(A') = 1
P(A U B) = P(A) + P(B) - P(AB)
Multiply through the branches of a tree to calculate probabilities of outcomes.
Add through the rows of the outcomes to calculate multiple options.
Construct a tree diagram showing all the ways a captain and a vice-captain can be selected from Matt, Rebecca, Helen and Chris.
Construct a tree diagram showing all possible outcomes of boys and girls a couple with three children can possibly have.
A coin is tossed twice.
Construct a tree diagram showing the results of the given experiment.
Use the tree diagram to find the probability of getting exactly 1 Head.
Use the tree diagram to find the probability of getting 2 heads.
Use the tree diagram to find the probability of getting no heads.
Use the tree diagram to find the probability of getting 1 Head and 1 Tail.
Investigate situations that involve elements of chance: A comparing discrete theoretical distributions and experimental distributions, appreciating the role of sample size B calculating probabilities in discrete situations.
Investigate a situation involving elements of chance