© Nuffield Foundation 2011

Nuffield Free-Standing Mathematics Activity

Laws of probability

© Nuffield Foundation 2010

When 2 dice are tossed, what is the probability that the sum of the scores is 11?

The laws of probability can help you to answer such questions.

For mutually exclusive events: P(A or B) = P(A) + P(B)

= P(Ace) + P(King)

Aces Kings

Venn diagram

Events A and B are mutually exclusive if A can occur or B can occur, but not both at the same time. A card drawn from a pack could be an Ace or King, but not both:

528

524P(Ace) 52

4P(King)

P(Ace or King)

131 13

1

132

Laws of probability

Laws of probability

If a card is drawn from a pack of 52:

524P(Ace)

Aces Hearts

Venn diagram

When events A and B are not mutually exclusive you cannot just add their probabilities.

5213P(Heart)

5216P(Ace or Heart)

Think aboutWhat would the Venn diagram look like?

5217not

Think aboutWhat is the probability of an ace or a heart?

Laws of probabilityEvents A and B are independent if neither has any effect on the probability of the other

For independent events:

261

131

21

P(A and B) = P(A) P(B)

P(Head and King)

131

131 P(2 Kings) 169

1

If a fair coin is tossed and a card drawn from a pack:

If 2 cards are drawn from a pack, for independence the 1st card must be replaced before the 2nd is drawn.

Biased coin with head twice as likely as tails

1st toss

H

T

2nd toss

H

T

H

T

Tree diagram

32

31

32

31

32

31

94

32

32 P(HH)

92

31

32 P(HT)

92

32

31 P(TH)

91

31

31 P(TT)

95

91

94 P(same result on both tosses) Total = 1

94

92

92 P(different results on the tosses) Total = 1

Think about What is the sum of these probabilities?

Reflect on your workExplain what is meant by the term ‘mutually exclusive’, and give an example.What is the law of probability that can be applied to mutually exclusive events?

What does the Venn Diagram of two mutually exclusive events look like?What does it look like if the events are not mutually exclusive?

Explain what is meant by the term ‘independent’, and give an example.What is the law of probability that can be applied to independent events?

Explain how the laws of probability are applied in a tree diagram.