Probability

AS Level Mathematics · Probability & Statistics 1. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

When all outcomes are equally likely, a probability is the number of outcomes you want divided by the total number of outcomes. The outcomes can be listed, or counted using permutations and combinations.

For combined events, multiply along the branches of a tree diagram ('and') and add the results of separate routes ('or'). Without replacement, the probabilities change on the second branch.

Two events are mutually exclusive if they cannot both happen. They are independent if one happening does not change the probability of the other. A conditional probability, P(A | B), is the probability of A given that B has happened: only the outcomes in B are now possible.

Rules to remember

Common mistake

Confusing mutually exclusive with independent. Mutually exclusive means P(A ∩ B) = 0; independent means P(A ∩ B) = P(A) × P(B). Test with the numbers given.

Worked example

Machine A makes 60% of the items in a factory and machine B makes 40%. 5% of the items from A are faulty and 10% of the items from B are faulty. An item chosen at random is faulty. Find the probability that it was made by A.

  1. P(A and faulty) = 0.6 × 0.05 = 0.03.
  2. P(B and faulty) = 0.4 × 0.10 = 0.04.
  3. P(faulty) = 0.03 + 0.04 = 0.07.
  4. P(A | faulty) = 0.03 ÷ 0.07 = 3/7 = 0.429 to 3 significant figures.

Answer: 3/7 (0.429)

Practice questions

Try each one, then open the answer.

1. A bag contains 5 red and 3 blue counters. Two counters are taken at random without replacement. Find the probability that exactly one is red.

  1. A
    15/28
  2. B
    15/32
  3. C
    15/56
  4. D
    5/14
Show answer

Answer: A. Red then blue: 5/8 × 3/7 = 15/56. Blue then red: 3/8 × 5/7 = 15/56. Add: 30/56 = 15/28. Forgetting the second order gives 15/56.

2. Three fair coins are tossed. Find the probability of obtaining exactly two heads.

  1. A
    1/8
  2. B
    1/4
  3. C
    3/8
  4. D
    1/2
Show answer

Answer: C. The 8 equally likely outcomes include HHT, HTH and THH with exactly two heads, so P = 3/8.

3. Events A and B have P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2. Which statement is correct?

  1. A
    A and B are mutually exclusive
  2. B
    P(A ∪ B) = 0.9
  3. C
    A and B are independent
  4. D
    P(A | B) = 0.4
Show answer

Answer: C. P(A) × P(B) = 0.5 × 0.4 = 0.2 = P(A ∩ B), so A and B are independent. They are not mutually exclusive since P(A ∩ B) ≠ 0, and P(A ∪ B) = 0.5 + 0.4 − 0.2 = 0.7.

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