4.2.3 Tree Diagram
A tree diagram is a visual representation of the possible outcomes of an event and their associated probabilities.
It is a useful tool for calculating the probability of complex events that involve multiple stages or components.
The diagram is composed of nodes and branches that represent the different possible outcomes and the probabilities associated with each outcome, respectively.
To draw a probability tree diagram, follow these steps:
Start with a single node at the top of the page representing the initial event or decision.
Draw lines extending downward from the central node to represent the different possible outcomes of that event or decision.
Label each branch with the probability of the associated outcome.
Add additional nodes and branches as necessary to represent subsequent events or decisions and their associated probabilities.
Continue this process until you have represented all possible outcomes and associated probabilities.
Calculate the probability of each final outcome by multiplying the probabilities along the branches that lead to that outcome.
Review the probability tree diagram to ensure it accurately represents the event and associated probabilities.
Here’s an example of a simple probability tree diagram:
Suppose you are flipping a coin twice, and you want to know the probability of getting two heads.
In this example, the initial event is flipping a coin, and there are two possible outcomes: heads or tails, each with a probability of \( 0.5 \).
We then repeat this event, resulting in a total of four possible outcomes.
The probability of getting two heads is the probability of following the branch “H” on the first coin flip, and then following the branch “H” on the second coin flip, which is \( (0.5)\times (0.5) = 0.25 \)
Test yourself
Question 1:
A bag contains \( 15 \) red beads and \( 10 \) yellow beads.
Ariana picks a bead at random, records its color and replaces it in the bag.
She then picks another bead at random.
Find the probability that she picks two red beads.
[2]
Find the probability that she does not pick two red beads.
[2]
Question 2:
Tanya plants some seeds.
The probability that a seed will produce flowers is \( 0.8 \).
When a seed produces flowers, the probability that the flowers are red is \( 0.6 \) and the probability
that the flowers are yellow is \( 0.3 \).
Complete the tree diagram.
[2]
Find the probability that a seed chosen at random produces red flowers.
[2]
Tanya chooses a seed at random.
Find the probability that this seed does not produce yellow flowers.
[3]
Question 3:
\( 40 \) children were asked if they have a computer or a phone or both.
The Venn diagram shows the results.
A child is chosen at random from the children who have a computer.
Write down the probability that this child also has a phone.
[1]
Question 4:
Box \( A \) and box \( B \) each contain blue and green pens only.
Raphael picks a pen at random from box \( A \) and Paulo picks a pen at random from box \( B \).
The probability that Raphael picks a blue pen is \( \frac{2}{3} \).
The probability that both Raphael and Paulo picks a blue pen is \( \frac{8}{15} \).
Find the probability that Paulo picks a blue pen.
[2]
Find the probability that both Raphael and Paulo pick a green pen.
[3]
Question 5:
The probability that it will rain tomorrow is \( \frac{5}{8} \).
If it rains, the probability that Rafael walks to school is \( \frac{1}{6} \).
If it does not rain, the probability that Rafael walks to school is \( \frac{7}{10} \).
Complete the tree diagram.
[3]
Calculate the probability that it will rain tomorrow and Rafael walks to school.
[2]