1. © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Laws of probability

2. © 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.

3. 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: P(Ace) P(King) P(Ace or King) Laws of probability

4. If a card is drawn from a pack of 52: P(Ace) AcesHearts Venn diagram When events A and B are not mutually exclusive you cannot just add their probabilities. P(Heart) P(Ace or Heart) Think about What would the Venn diagram look like? not Think about What is the probability of an ace or a heart?

5. Laws of probability Events A and B are independent if neither has any effect on the probability of the other For independent events: P(A and B) = P(A)  P(B) P(Head and King) P(2 Kings) If a fair coin is tossed and a card drawn from a pack: If 2 cards are drawn from a pack, for independence the 1 st card must be replaced before the 2 nd is drawn.

6. Biased coin with head twice as likely as tails 1st toss H T 2nd toss H T H T Tree diagram P(HH) P(HT) P(TH) P(TT) P(same result on both tosses) Total = 1 P(different results on the tosses) Total = 1 Think about What is the sum of these probabilities?

7. Reflect on your work Explain 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.