Probability

Probability is the branch of mathematics that measures uncertainty. Weather forecasts and insurance depend on it. This chapter begins with a random experiment, which has more than one possible outcome and whose result cannot be predicted in advance. The set of all outcomes is the sample space S, and any subset of S is an event. You will study simple and compound events, sure and impossible events, the complement of an event, and events formed with 'and' and 'or'. Mutually exclusive events cannot occur together, while exhaustive events together cover the whole sample space. The axiomatic approach treats probability as a function assigning a number P(E) to every event, with 0 <= P(E) <= 1, P(S) = 1 and P(A or B) = P(A) + P(B) for mutually exclusive A and B. From these follow P(not A) = 1 - P(A) and the addition rule P(A or B) = P(A) + P(B) - P(A and B). For equally likely outcomes, P(E) = n(E)/n(S). These ideas are applied to coins, dice, cards and selections using combinations.

Detailed Worksheet: Probability Section A - Definitions (10 marks) 1. Define a random experiment and a sample space with one example. (2 marks) 2. Write the sample space when two coins are tossed and find the probability of at least one head. (2 marks) 3. A die is thrown. Find the probability of getting a prime number. (2 marks) 4. If P(A) = 2/11, find P(not A). (2 marks) 5. Distinguish between mutually exclusive and exhaustive events. (2 marks) Section B - Calculations and Applications (15 marks) 6. A card is drawn from a well-shuffled pack of 52 cards. Find the probability that it is a king or a heart. (3 marks) 7. If P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16, find P(A or B), P(not A) and P(not B). (3 marks) 8. Two dice are thrown. Find the probability that the sum is 8. (3 marks) 9. A committee of 2 is chosen from 4 men and 6 women. Find the probability that both are women. (3 marks) 10. Three coins are tossed. Find the probability of exactly two heads and of at least two heads. (3 marks) Section C - Diagrams (10 marks) 11. Draw a Venn diagram for two events A and B and shade the region for A or B. Write the addition rule. (4 marks) 12. Draw a 6 x 6 table of outcomes when two dice are thrown and mark the outcomes whose sum is 7. Find the probability. (3 marks) 13. Draw a tree diagram for tossing a coin three times and list the sample space. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. State the axioms of probability. Using them, prove that P(not A) = 1 - P(A) and that P(A or B) = P(A) + P(B) - P(A and B). (5 marks) 15. Four cards are drawn from a pack of 52 cards. Find the probability of getting 3 diamonds and 1 spade. (5 marks) 16. Two dice are thrown. A is the event that the sum is even and B the event that the sum is a multiple of 3. Find P(A), P(B), P(A and B) and P(A or B). (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Give answers as fractions in their lowest terms or as decimals.