Probability

Probability measures how likely an event is, and this chapter extends the ideas of Class 11 to situations where some information is already known. You will learn conditional probability, P(E/F) = P(E and F)/P(F), which gives the probability of E when F has already occurred, and its properties. From it follows the multiplication theorem, P(E and F) = P(F) P(E/F), used to find the probability of events happening together, such as drawing two cards in succession without replacement. You will learn when two events are independent, so that P(E and F) = P(E) P(F). You will then study the theorem of total probability for a partition of the sample space and Bayes' theorem, which reverses conditional probabilities to find the probability of a cause given an observed effect, as in medical tests, defective items from different machines and insurance claims. Finally, you will learn about random variables and their probability distributions, and calculate the mean or expectation of a random variable, a key idea in statistics and decision making.

Detailed Worksheet: Probability Section A - Definitions (10 marks) 1. Define the conditional probability of an event E given that F has occurred. State one property of conditional probability. (2 marks) 2. When are two events E and F said to be independent? Are mutually exclusive events with non-zero probabilities independent? (2 marks) 3. State the multiplication theorem of probability for two events. (2 marks) 4. What is a partition of a sample space? State the theorem of total probability. (2 marks) 5. Define a random variable and its probability distribution. Give one example. (2 marks) Section B - Calculations and Applications (15 marks) 6. If P(E) = 0.6, P(F) = 0.3 and P(E and F) = 0.2, find P(E/F) and P(F/E). (3 marks) 7. A family has two children. Find the probability that both are boys, given that at least one of them is a boy. (3 marks) 8. Events A and B are independent with P(A) = 0.3 and P(B) = 0.4. Find (i) P(A and B), (ii) P(A or B) and (iii) P(neither A nor B). (3 marks) 9. Bag I contains 3 red and 4 black balls and bag II contains 5 red and 6 black balls. One bag is chosen at random and a ball drawn from it is red. Find the probability that it was drawn from bag II. (3 marks) 10. A coin is tossed twice. Let X be the number of heads. Write the probability distribution of X and find its mean. (3 marks) Section C - Diagrams (10 marks) 11. Draw a tree diagram for question 9, showing the probabilities of choosing each bag and of drawing a red or black ball, and use it to find the total probability of drawing a red ball. (4 marks) 12. Draw a Venn diagram for two events E and F and shade the region representing E and F. Use it to explain the formula for P(E/F). (3 marks) 13. Draw a bar diagram for the probability distribution of the number of heads in three tosses of a coin. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. In a factory, machines M1, M2 and M3 produce 25%, 35% and 40% of the bolts, of which 5%, 4% and 2% respectively are defective. A bolt is drawn at random and found defective. Find the probability that it was made by M2. (5 marks) 15. A disease affects 0.1% of a population. A test detects the disease in 99% of people who have it, but also gives a positive result for 0.5% of healthy people. Find the probability that a person who tests positive actually has the disease. Comment on why the answer is surprisingly low. (5 marks) 16. A random variable X takes the values 0, 1, 2, 3 and 4 with probabilities 0.1, k, 2k, 2k and k respectively. Find k, P(X >= 2) and the mean of X. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Define events clearly before applying formulas and give answers as fractions in lowest terms.