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Lesson 11 of 13
  • Year 9

Calculating theoretical probabilities from probability trees (two events)

I can calculate and use theoretical probabilities for combined events using probability trees (2 events).

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Lesson 11 of 13
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  • Year 9

Calculating theoretical probabilities from probability trees (two events)

I can calculate and use theoretical probabilities for combined events using probability trees (2 events).

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Share activities with pupils

These resources will be removed by end of Summer Term 2025.

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These resources were created for remote use during the pandemic and are not designed for classroom teaching.

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Lesson details

Key learning points

  1. The probability of an outcome can be found with a probability tree diagram showing all possible outcomes for two events.
  2. The probability of a set of outcomes can be found using a probability tree diagram.
  3. The probability of a set of outcomes can be found using a probability tree, even when outcomes are not equally likely.

Keywords

  • Outcome - An outcome is a result of a trial (e.g. getting heads when flipping a coin once or getting two heads when flipping a coin twice).

  • Event - An event is a subset of a sample space. i.e. An outcome or set of outcomes that may occur from a trial (e.g. flipping a coin twice and getting the same result each time).

  • Probability tree - Each branch of a probability tree shows a possible outcome from an event or from a stage of a trial, along with the probability of that outcome happening.

  • Sample space - A sample space is all the possible outcomes of a trial. A sample space diagram is a systematic way of producing a sample space.

Common misconception

When I see a probability tree, I always multiply the probabilities.

If a probability tree shows a two-stage trial, then the probability of an outcome is the product of the probabilities of the outcomes at each stage. The probability of an event is the sum of the probabilities of each outcome in the event.


To help you plan your year 9 maths lesson on: Calculating theoretical probabilities from probability trees (two events), download all teaching resources for free and adapt to suit your pupils' needs...

Two-layer probability trees can be used to show the probabilities of outcomes from a two-stage trial (with each stage represented as a layer of branches), as well as the probabilities of two events from a trial (with each event represented by a layer of branches).
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This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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Prior knowledge starter quiz

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6 Questions

Q1.
What is the sum of the possible events A, B and C in this probability tree?

An image in a quiz
Correct answer: 1
$$/frac{7}{8}$$
$$/frac{1}{8}$$
$$/frac{2}{8}$$

Q2.
In a fair six-sided die, what is the probability of rolling a 2 and getting 'tails' in a single toss of a fair coin?

Correct answer: $$\frac{1}{12}$$
$$\frac{1}{6}$$
$$\frac{1}{2}$$

Q3.
There are two trials. In Trial 1, a spinner with {A, B} is spun twice. In Trial 2, a spinner with {B, C} is spun twice. Which statements are true?

Correct answer: A and C have an equal chance of appearing in the trials.
Correct answer: B is more likely to appear in the trials.
B is less likely to appear in the trials than A.

Q4.
What is the probability of this spinner landing on a vowel?

An image in a quiz
Correct answer: 0.4
0.2
0.5
0.6

Q5.
Alex believes that the likelihood of a spinner stopping on a shaded space is equal to an unshaded space. Is Alex correct?

An image in a quiz
Correct answer: Alex is correct.
Alex is incorrect, it is less likely the spinner would land on a shaded space.
Alex is incorrect, it is more likely the spinner would land on a shaded space.

Q6.
How many ways are there to get a sum of 7 when adding the results of two standard six-sided dice rolls?

5
Correct answer: 6
7

Assessment exit quiz

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6 Questions

Q1.
If P(A) = $$\frac{1}{3}$$ and P(A and D) = $$\frac{1}{5}$$, then what is P(D)?

An image in a quiz
Correct answer: $$\frac{3}{5}$$
$$\frac{1}{5}$$
$$\frac{2}{5}$$

Q2.
What is the error in this tree diagram?

An image in a quiz
Events B and F should sum to 1 but they don't.
Events A and C should sum to 1 but they don't.
Events D and E should sum to 1 but they don't.
Correct answer: Events E and F should sum to 1 but they don't.

Q3.
What is a suitable calculation for the probability of event B occurring, and event F not occurring?

An image in a quiz
Correct answer: $$\frac{5}{6}*\frac{1}{4}$$
$$\frac{5}{6}+\frac{1}{4}$$
$$\frac{5}{6}*\frac{3}{4}$$
$$\frac{5}{6}+\frac{3}{4}$$

Q4.
If the first and second event in this tree diagram were both flipping a coin once, what would be the probability of event B and event F occurring?

An image in a quiz
Correct answer: $$\frac{1}{4}$$
$$\frac{1}{2}$$
$$\frac{2}{6}$$

Q5.
What is the least likely (but not impossible) pair of events occurring in this tree diagram?

An image in a quiz
Correct answer: A then C
A then D
B then E
B then F

Q6.
Which pair of events has a more than even chance of occurring?

An image in a quiz
Correct answer: B and F
B and E
A and C
A d D