Statistics: Probability
Probability is a measure, between 0 and 1, of how likely an event is to occur, and A Level Statistics builds on GCSE probability rules using formulae for combined and conditional events. This includes the addition rule P(A or B) = P(A) + P(B) - P(A and B), the multiplication rule for independent events, and conditional probability P(A|B) = P(A and B)/P(B).
Method
- Use the addition rule P(A or B) = P(A) + P(B) - P(A and B) to find the probability that at least one of two events occurs.
- Use P(A and B) = P(A) x P(B) to test for independence; two events are independent only if this equation holds (or equivalently, P(A|B) = P(A)).
- Use conditional probability P(A|B) = P(A and B) / P(B) when a question restricts attention to outcomes where B has already happened.
- For without-replacement problems, draw a tree diagram (or work step by step) so that probabilities on later branches change to reflect what has already been removed.
- Use the binomial formula P(X = r) = nCr x p^r x (1-p)^(n-r) for a fixed number of independent trials with two outcomes and a constant probability of success.
- Use the law of total probability, P(A) = P(B)P(A|B) + P(B')P(A|B'), to combine information from two or more branches, and reverse this to answer 'given that ... occurred, find the probability it came from ...' questions.
Worked example
A bag contains 6 red counters and 4 blue counters. Two counters are drawn at random, one after another, without replacement. Find the probability that both counters drawn are red.
- P(first counter red) = 6/10.
- Given the first counter drawn was red, 5 red and 4 blue counters remain, so P(second red | first red) = 5/9.
- P(both red) = P(first red) x P(second red | first red) = (6/10) x (5/9).
- (6/10) x (5/9) = 30/90 = 1/3.
Practice questions
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Q1For events A and B, P(A) = 0.5 and P(B) = 0.3. Given that A and B are mutually exclusive, find P(A or B).Show answer
Answer: 0.8 (P(A) + P(B), since P(A and B) = 0 for mutually exclusive events).
Q2For events A and B, P(A) = 0.4, P(B) = 0.5 and P(A and B) = 0.2. Find P(A or B).Show answer
Answer: 0.7 (0.4 + 0.5 - 0.2).
Q3For events A and B, P(A) = 0.6, P(B) = 0.25 and P(A and B) = 0.15. Determine whether A and B are independent.Show answer
Answer: Yes; P(A) x P(B) = 0.6 x 0.25 = 0.15, which equals P(A and B).
Q4A box contains 8 pens: 3 blue and 5 black. Two pens are chosen at random without replacement. Find the probability that both are blue.Show answer
Answer: 3/28 (= (3/8) x (2/7)).
Q5For events C and D, P(C) = 0.45, P(D) = 0.3 and P(C|D) = 0.5. Find P(C and D).Show answer
Answer: 0.15 (P(C|D) x P(D) = 0.5 x 0.3).
Q6A factory uses two machines. Machine X produces 70% of items and Machine Y produces 30%. Machine X's items are defective 2% of the time, and Machine Y's items are defective 6% of the time. Find the probability that a randomly chosen item is defective.Show answer
Answer: 0.032 (0.7 x 0.02 + 0.3 x 0.06 = 0.014 + 0.018).
Exam-style questions
Written in the style of a A Level Maths exam paper, with a full mark scheme.
For events A and B, P(A) = 0.55, P(B) = 0.4 and P(A and B) = 0.2. Find P(A or B), giving your answer as a decimal.
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A bag contains 9 counters: 4 green and 5 yellow. Two counters are drawn at random, one after another, without replacement. (a) Find the probability that both counters are the same colour. (3) (b) Given that both counters drawn are the same colour, find the probability that both are green. (2)
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A survey of 200 people found that C is the event a randomly chosen person owns a car and B is the event a randomly chosen person owns a bicycle. It is known that P(C) = 0.6, P(B) = 0.35 and P(neither a car nor a bicycle) = 0.15. (a) Show that P(C and B) = 0.10. (2) (b) Find the number of the 200 people who own a car but not a bicycle. (1) (c) Two people are selected at random, without replacement, from those who own a bicycle. Find the probability that both of them also own a car. (3)
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Free printable worksheet
Want more practice on paper? Download the statistics: probability 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.
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