Suggest — 2 marks
A student is investigating the reliability of a new type of light bulb. They test two bulbs independently. Each bulb has a probability of 0.8 of lasting longer than 1000 hours. The student wants to use a tree diagram to represent all possible outcomes.
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(a) Suggest what the two main branches of the tree diagram should represent.
[1 mark]
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(b) Suggest the probability that should be written on a branch representing a bulb that does NOT last longer than 1000 hours.
[1 mark]
Show mark scheme
- (a) The two branches should represent the outcomes for the first bulb (or first event) / bulb 1 lasting longer than 1000 hours OR not lasting longer than 1000 hours. Accept: 'success and failure' or 'works and doesn't work' or equivalent outcomes for the first bulb.
- (b) 0.2 (or 1 − 0.8 or 20% or 1/5). Accept equivalent fractions or percentages.
Define — 3 marks
A quality control manager at a manufacturing plant tests electronic components for defects. She randomly selects components and tests them twice independently. Each test has a probability of correctly identifying a defective component. The manager uses a tree diagram to calculate the probability that a component will pass both tests.
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(a) Define what is meant by independent events in the context of probability.
[1 mark]
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(b) Define the term 'combined event' and explain how tree diagrams are used to represent combined events.
[2 marks]
Show mark scheme
- (a) Award 1 mark for: Events where the outcome of one event does not affect the probability of the other event occurring / the result of the first test does not influence the result of the second test.
- (b) Award 1 mark for: A combined event is when two or more separate events occur together / the result of two or more events considered simultaneously. Award 1 mark for: Tree diagrams show all possible outcomes by branching to represent each stage/event, allowing probabilities to be multiplied along branches to find the probability of combined events.
Calculate — 2 marks
A bag contains 5 red counters and 3 blue counters. A counter is taken at random from the bag, its colour is recorded, and then it is replaced. A second counter is then taken at random from the bag.
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(a) Calculate the probability that both counters are red.
[1 mark]
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(b) Calculate the probability that the two counters are different colours.
[1 mark]
Show mark scheme
- (a) 5/8 × 5/8 or equivalent method
- (a) 25/64 or 0.390625 or 39.0625%
- (b) 5/8 × 3/8 + 3/8 × 5/8 or equivalent method
- (b) 30/64 = 15/32 or 0.46875 or 46.875%
Show — 3 marks
A bag contains 4 green counters and 6 yellow counters. Mia takes a counter at random from the bag, notes its colour, and replaces it. She then takes a second counter at random. The tree diagram shows the possible outcomes.
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(a) Write down the probability that Mia takes a green counter on her first draw.
[1 mark]
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(b) Show that the probability of Mia taking two counters of different colours is 12/25.
[2 marks]
Show mark scheme
- (a) 4/10 oe
- (b) Correct method: (4/10 × 6/10) + (6/10 × 4/10) oe
- (b) Complete working shown leading to 12/25 oe (e.g., 24/100 + 24/100 = 48/100 = 12/25)
Explain — 2 marks
A fair spinner has 4 equal sections numbered 1, 2, 3 and 4. The spinner is spun twice. The tree diagram below shows all possible outcomes.
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(a) Write down the probability of getting a 3 on the first spin.
[1 mark]
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(b) Explain why there are 16 possible outcomes shown on the tree diagram.
[1 mark]
Show mark scheme
- (a) 1/4 or 0.25
- (b) Because there are 4 outcomes for the first spin AND 4 outcomes for the second spin, and 4 × 4 = 16 (or equivalent)
GCSE Perfect