Probability

NUST NET (Engineering) · Maths · Permutations, combinations and probability. 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, the probability of an event is the number of favourable outcomes divided by the total number of outcomes. It always lies between 0 and 1, and an event and its complement add up to 1.

The addition rule is for "A or B". If the two events can happen together, the overlap must be subtracted once. The multiplication rule is for "A and B". For independent events simply multiply; when items are drawn without replacement, the second probability changes because one item has gone.

For "at least one", it is nearly always quicker to find the probability of none and subtract it from 1.

Rules to remember

Common mistake

Adding P(A) and P(B) for "or" without subtracting the overlap. Kings or hearts is 4/52 + 13/52 − 1/52 = 16/52, because the king of hearts would otherwise be counted twice.

Worked example

A bag contains 4 red and 6 blue balls. Two balls are drawn one after the other without replacement. The probability that they are of different colours is

  1. Red then blue: (4/10) × (6/9) = 24/90.
  2. Blue then red: (6/10) × (4/9) = 24/90.
  3. These two cases are separate, so add: 24/90 + 24/90 = 48/90 = 8/15.

Answer: 8/15

Practice questions

Try each one, then open the answer.

1. A card is drawn at random from a pack of 52 playing cards. The probability that it is a face card (J, Q or K) or a red card is

  1. A
    19/26
  2. B
    3/13
  3. C
    8/13
  4. D
    3/26
Show answer

Answer: C. There are 12 face cards, 26 red cards and 6 red face cards, so the favourable count is 12 + 26 − 6 = 32 and P = 32/52 = 8/13. Not removing the 6 counted twice gives 38/52 = 19/26.

2. For events A and B, P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2. The probability that neither A nor B occurs is

  1. A
    0.1
  2. B
    0.7
  3. C
    0.8
  4. D
    0.3
Show answer

Answer: D. P(A ∪ B) = 0.5 + 0.4 − 0.2 = 0.7, so P(neither) = 1 − 0.7 = 0.3. Forgetting to subtract P(A ∩ B) gives 1 − 0.9 = 0.1.

3. A bag contains 5 red and 3 green balls. Two balls are drawn one after the other without replacement. The probability that both are green is

  1. A
    9/64
  2. B
    3/28
  3. C
    5/14
  4. D
    15/56
Show answer

Answer: B. P = (3/8) × (2/7) = 6/56 = 3/28, because after one green ball is taken only 2 green remain among 7 balls. (3/8)2 = 9/64 would be the answer with replacement.

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