Tree Diagrams for Independent Events
A probability tree diagram is a branching diagram showing the probabilities of two or more independent events happening one after another, such as spinning a spinner twice, with each branch labelled with an outcome and its probability, and every set of branches leaving the same point adding up to 1. The probability of one specific sequence of outcomes is found by multiplying the probabilities along that path, and the probability of reaching any one of several outcomes is found by adding the separate path probabilities together.
Before you start
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Method
- Draw two (or more) branches from a starting point for the first event, and label each branch with its outcome and its probability; the probabilities on branches from the same point must add up to 1.
- From the end of each first-event branch, draw a further set of branches for the second event, again labelling each with its outcome and probability; for independent events, these probabilities are the same regardless of which first-event branch they follow.
- To find the probability of reaching one specific end point (one particular combination of outcomes), multiply the probabilities along the branches that lead to it.
- To find the probability of one of several different end points happening (an 'or' situation, such as exactly one success), find the probability of each qualifying end point separately, then add these probabilities together.
- Before multiplying, check the two events really are independent, meaning the outcome of the first event does not change the probabilities for the second event.
- As a check, the probabilities of all of the end points in a completed tree diagram (every possible route through the tree) must add up to 1.
- Convert word problems into a labelled tree by first identifying the two events and their outcomes, writing clear branch labels before filling in the probabilities.
Worked example
A spinner is split into two sections and is spun twice. On each spin, the probability of landing on Blue is 0.3, and the probability of landing on Red is 0.7. Draw a tree diagram for two spins and use it to find the probability of landing on Blue exactly once in the two spins.
- Draw two branches for spin 1: Blue with probability 0.3, and Red with probability 0.7.
- From each of these, draw two further branches for spin 2, using the same probabilities again since the spins are independent: Blue (0.3) and Red (0.7).
- 'Exactly one Blue' happens along two different routes through the tree: Blue then Red, or Red then Blue.
- Multiply along each of these two routes: P(Blue, Red) = 0.3 x 0.7 = 0.21, and P(Red, Blue) = 0.7 x 0.3 = 0.21.
- Add the two probabilities together, since either route gives exactly one Blue: 0.21 + 0.21 = 0.42.
Practice questions
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Q1A fair six-sided dice is rolled twice. Work out the probability of rolling a six both times.Show answer
Answer: 1/36 (1/6 x 1/6)
Q2A biased spinner has P(Red) = 0.2 on each independent spin. The spinner is spun twice. Work out the probability of getting AT LEAST ONE red.Show answer
Answer: 0.36 (1 - 0.8 x 0.8 = 1 - 0.64)
Q3A biased coin has P(Heads) = 0.6 and P(Tails) = 0.4. The coin is flipped twice. Work out the probability of getting two tails.Show answer
Answer: 0.16 (0.4 x 0.4)
Q4A spinner has P(Win) = 0.25 and P(Lose) = 0.75 on each spin. The spinner is spun twice, and the spins are independent. Work out the probability of winning on both spins.Show answer
Answer: 0.0625 (1/16)
Q5Bag A contains counters such that P(red) = 0.3. A counter is drawn from Bag A, replaced, and then a second counter is drawn from Bag A. Work out the probability that both counters drawn are NOT red.Show answer
Answer: 0.49 (0.7 x 0.7)
Q6A basketball player has a probability of 0.75 of scoring each free throw, and each attempt is independent. She takes two free throws. Work out the probability that she scores on the first throw but misses the second.Show answer
Answer: 0.1875 (0.75 x 0.25)
Q7Explain why the probabilities on a pair of branches leaving the same point on a tree diagram must always add up to 1.Show answer
Answer: The branches leaving one point represent every possible outcome of that event, and since something must happen, the probabilities of all the possible outcomes of a single event must add up to 1 (certainty).
Q8A machine independently produces two components. The probability that each component is faulty is 0.05. Work out the probability that at least one of the two components is faulty.Show answer
Answer: 0.0975 (1 - 0.95 x 0.95 = 1 - 0.9025)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A box contains a large number of pens. Each pen is independently either Blue or Black. The probability that a pen is Blue is 0.35. Nadia picks two pens from the box (assume each pick is independent). (a) Draw a tree diagram to show the possible outcomes and their probabilities. (b) Find the probability that Nadia picks two pens of the SAME colour.
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A quiz has three independent multiple-choice questions. For each question, the probability that Leon guesses the correct answer is 1/4. Find the probability that Leon gets exactly one question correct out of the three.
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Two traffic lights on Freya's route to school operate independently. The probability that the first light is red when she arrives is 0.4, and the probability that the second light is red when she arrives is 0.5. Find the probability that AT LEAST ONE of the two lights is red.
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Free printable worksheet
Want more practice on paper? Download the tree diagrams for independent events worksheet pack - 7 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 27 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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