Probability

Words like "probably", "most likely" and "no chance" express uncertainty in everyday speech. Probability turns these words into numbers. In Class 9 you met experimental probability, found from the results of repeated trials. In this chapter you will study theoretical (classical) probability, which assumes that all outcomes of an experiment are equally likely. The probability of an event E is P(E) = (number of outcomes favourable to E) / (number of all possible outcomes). You will see that the probability of any event lies between 0 and 1, that an impossible event has probability 0 and a sure event has probability 1, and that the sum of the probabilities of all elementary events is 1. The event "not E" is the complement of E, and P(E) + P(not E) = 1. You will apply these ideas to tossing coins, throwing one or two dice, drawing cards from a well-shuffled deck of 52 cards and drawing balls from a bag, and to real situations such as lucky draws, defective items and games at a fair.

Detailed Worksheet: Probability Section A - Definitions (10 marks) 1. Define an experiment, an outcome and an event, giving one example of each. (2 marks) 2. What are equally likely outcomes? Give one example of an experiment whose outcomes are equally likely and one whose outcomes are not. (2 marks) 3. State the formula for the theoretical probability of an event E. (2 marks) 4. Define a sure event and an impossible event. Write the probability of each. (2 marks) 5. What are complementary events? State the relation between P(E) and P(not E). (2 marks) Section B - Calculations and Applications (15 marks) 6. A bag contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out at random. Find the probability that it is (i) red (ii) white (iii) not green. (3 marks) 7. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (i) a king of red colour (ii) a face card (iii) a spade. (3 marks) 8. Two dice are thrown at the same time. Find the probability that (i) the sum is 8 (ii) both show the same number (iii) the sum is 13. (3 marks) 9. A box contains 90 discs numbered from 1 to 90. One disc is drawn at random. Find the probability that it bears (i) a two-digit number (ii) a perfect square number (iii) a number divisible by 5. (3 marks) 10. A piggy bank contains hundred 50 p coins, fifty Rs 1 coins, twenty Rs 2 coins and ten Rs 5 coins. If one coin falls out when the bank is turned upside down, find the probability that the coin (i) will be a 50 p coin (ii) will not be a Rs 5 coin. (3 marks) Section C - Diagrams (10 marks) 11. Draw the sample space of throwing two dice together as a 6 by 6 grid of ordered pairs (write the pairs neatly in rows). Circle the outcomes for which the sum is 7 and find the probability of getting a sum of 7. (4 marks) 12. Draw a tree diagram for tossing three coins one after another, showing all 8 outcomes. Using it, find the probability of getting exactly two heads. (3 marks) 13. A game of chance consists of spinning an arrow that comes to rest pointing at one of the numbers 1 to 8 on a circular board divided into 8 equal sectors. Draw the board and find the probability that the arrow points at (i) an odd number (ii) a number greater than 2 (iii) a number less than 9. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A student argues that there are 11 possible outcomes 2, 3, 4, ..., 12 when two dice are thrown, so each has probability 1/11. Do you agree? Give reasons, and find the correct probability of getting a sum of 2 and a sum of 7. (5 marks) 15. Hanif tosses three coins together. He wins if all three show heads or all three show tails, and loses otherwise. Find the probability that Hanif will lose. A friend says that if you toss two coins, the outcomes two heads, two tails and one of each are equally likely, each with probability 1/3. Is this correct? Justify. (5 marks) 16. Explain why the probability of an event can never be negative or greater than 1. Which of the following cannot be the probability of an event: 2/3, -1.5, 15%, 0.7, 1.3? If P(E) = 0.05, what is the probability of not E? (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Write the total number of outcomes and the number of favourable outcomes before calculating each probability. Give answers as fractions in lowest terms.