Worksheet 4 th Topic : PROBABILITY TIME : 4 X 45 minutes STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving. BASIC COMPETENCY: 1.6 To determine possible number of outcomes of an event. 1.7 To determine the probability of an event and its interpretation. In this chapter, you will learn about: Classical Definition of Probability Expectation Frequency F. PROBABILITY If (A) n denotes the number of outcomes in the event A and n (S) the number of a possible outcomes in the sample space, then probability of the event A is P ( A) = n( A) n( S) For any event A, 0 P ( A) 1. If P ( A) = 0,then the event cannot possibly occur. If P ( A) = 1, then the event will certainly occur. Example 27 A die is thrown. Find the probability of the event of getting prime numbers? The sample space S = {.. } The outcomes in the event A = {..} n (A) = 28
n (S) = P (A) = Example 28 A card is drawn from a pack of 52 playing cards (well-shuffled so that the drawing is random) a. What is the total number of possible outcomes of this experiment? b. How many of these outcomes have the occurrence of i. a black card, ii. a red ace, iii. a diamond, iv. a card which is not a diamond? Write down the probability of each event. a. n (S) = b. There are black cards in the pack. P(a black card) = =... c. There are red aces. P(a red ace) = =... d. There are diamonds in the pack. P(a diamond) = =... e. There are diamonds in the pack, the remaining cards in the pack are not diamond. P(not a diamond) = =... Example 29 Two dice are thrown together. Find the probability that the sum of the resulting numbers is a. odd, b. even, c. a prime number, d. a multiple of 4, e. at least 7. Second die First die + 1 2 3 4 5 6 1 (2,1) 2 3 4 (1,4) 5 6 A possibility diagram There are possible outcomes in the sample space S, so n (S) = We define the following events: 29
A : the sum is odd. B : the sum is even. C : the sum is a prime number D : the sum is a multiple of 4. E : the sum is at least 7. a. From the possibility diagram, A = {(1,2), (1,4), (1, ), (2, ), (2, ), (2, ), (3, ), (3, ), (3, ), (4, ), (4, ), (4, ), (5, ), (5, ), (5, ), (6, ), (6, ), (6, )} n (A) = P ( A) = =... b. The sum is even. n (B) = P ( B) = =... c. n (C) = P ( C) = =... d. n (D) = P ( D) = =... e. n (B) = P ( B) = =... A and B are called complementary events. B is said to be complementary to A ( B = A' and A = B' ). We note that P ( A) + P( B) = 1, so P( A) = 1 P( A' ) Example 30 A box contains 5 red, 3 yellow, and 2 blue discs. Two discs are drawn at random from the box one after another. a. What is the probability that the first disc drawn will be red? b. If the first disc drawn is blue and it is not replaced, what is the probability of drawing a yellow disc on the second draw? a. P(the first disc drawn is red) = P(r r, r y, r ) =.. b. P(... ) = 30
Example 31 A box contains 10 balls: 5 red, 3 white, and 2 blue. Three balls are drawn at random from the box. What is the probability the three balls will be a. all of red, b. two red and one white, c. different colours? n(s) =. a. b. c. Expectation Frequency In general if an experiment will be done N times and the probability of an event = P (A), then the multiplication of N P(A) is the expectation frequency Expectation Frequency = N P(A) Example 32 A coin is thrown 100 times. How many the expectation frequency of the event? 31
Example 33 Two dice are thrown together 360 times. How many the expectation frequency of the sum of the resulting numbers is? a. even, b. a prime number, c. a multiple of 4, d. at least 7. Conditional Probability If E 1 and E 2 are two events, the probability that E 2 occurs given that E 1 has occurred is denoted by P(E 2 E 1 ). P(E 2 E 1 ) is called the conditional probability of E 2 given that E 1 has occurred. Calculating Conditional Probability Let E 1 and E 2 be any two events defined in a sample space S such that P(E 1 ) > 0. The conditional probability of E 2, assuming E 1 has already occurred, is given by ( E E ) P = 2 1 P( E2 and E1 ) P ( E ) 1 Example 34 Let A denote the event `student is female' and let B denote the event `student is French'. In a class of 100 students suppose 60 are French, and suppose that 10 of the French students are females. Find the probability that if I pick a French student, it will be a girl, that is, find P(A B). 32
Example 35 What is the probability that the total of two dice will be greater than 8, given that the first die is a 6? Exercise 7 1. A large fish tank contains 36 goldfish of which 8 are red and 28 are black. One goldfish is picked at random. a. What is the probability that it will be red? b. Assuming that a red goldfish is picked the first time and put outside, what is the probability that the second goldfish picked will be red? 2. All the clubs are removed from a pack of ordinary playing cards. a. A card is drawn at random from the remaining cards in the pack. Find the probability of drawing (i) a red card (ii) a heart (iii) a picture card (iv) a card that is not an ace b. Assuming that a jack of spade is drawn the first time and put aside, find the probability that the second card drawn is (i) a black card (ii) an ace (iii) not a picture card 3. In a class of 30 pupils, 12 are girls and two of them are short-sighted. Among the 18 boys, 6 are short-sighted. If a pupil is selected at random, what is the probability that the pupil chosen will be a. a girl, b. short-sighted, c. a short-sighted boy? 4. Each of the letters of the word MATHEMATICS is written on a card. All the eleven cards are well-shuffled and placed down on a table. If a card is turned over, what is the probability that the card bears a. the letter M, b. vowel, c. the letter P. 33
5. A two-digit number is written down at random. Find the probability that the number will be a. smaller than 20 b. even c. a multiple of 5 6. A box of 2 dozen pencils contains 8 pencils with broken points. What is the probability of picking one pencil without a broken point? 7. Peter has 10 books, 5 Chinese books and 5 English books, in his bag. Three of his ten books are science fiction which are in English. If a book is chosen at random, find the probability of choosing a. an English book, b. a Malay book, c. either a Chinese book or an English book, d. a science fiction book, e. an English book which is not a science fiction book, f. a book which is not a science fiction book. 8. A bag contains 15 balls of which x are red. Write an expression for the probability that a ball drawn at random from the bag is red. When 5 more red balls are added to the bag, the probability become ¾. Find the value of x. 9. The figure below shows a circle devided into sectors of different colours. If a point is selected at random in the circle, calculate the probability that it lies in the a. yellow sector b. red sector blue c. blue sector d. green or white sector yellow green 60 0 30 0 white 45 0 135 0 red 10. A class has 16 boys and 24 girls. Of the 16 boys, 3 are left-handed and of the 24 girls, 2 are left-handed. a. Mrs. Ani, the form teacher, selects a pupil to run an errand. Assuming that she is equally likely to select a pupil, what is the probability that she selects (i) a boy (ii) a left-handed pupil b. While waiting a pupil to return, she needs another pupil to clean the blackboard. She selects one of the remaining pupils at random. Assuming that the first pupil she sent away is a girl who is not left-handed, find the probability that she selects (i) a left-handed girl (ii) a left-handed boy 11. A die is first thrown and a coin is then tossed. List all the possible outcomes of the sample space of the experiment using possibility diagram. With the help of the possibility diagram, find the probability of the events A, B, C, and D as defined below. A: getting a prime number on the die and the coin showing head. B: getting an even number on the die and the coin showing tail. C: getting a number greater than 4 on the die. D: getting a number less than 5 on the die. 34
12. A box contains three cards bearing the numbers 1, 2, and 3. A second box contains four cards bearing the number 2, 3, 4, and 5. A card is chosen at random from each box. a. Display all the possible outcomes of the experiment using a possibility diagram. b. With the help of the possibility diagram, calculate the probability that (i) the cards bear the same number, (ii) the numbers on the cards are different, (iii) the larger of the two numbers on the card is 3, (iv) the sum of the two numbers on the cards is less than 7, (v) the product of the two numbers on the cards is at least 8. 13. In a game, the player throws a coin and a six-faced die simultaneously. If the coin shows a head, the player s score is the score on the die. If the coin shows a tail, then the player s score is twice the score on the die. Some of the player s possible scores are shown in the possibility diagram given. Coin Die 1 2 3 4 5 6 H T 6 a. Copy and complete the possibility diagram. b. Using the diagram, find the probability that the player s score is (i) odd (ii) even (iii) a prime number (iv) less than or equal to 8 (v) a multiple of 3 14. Two six-sided dice were thrown together and the difference of the resulting numbers on their faces was calculated. Some of the differences are shown in the possibility diagram given. 1 st die 1 2 3 4 5 6 1 0 2 1 4 3 4 5 6 4 0 2 nd die a. Copy and complete the possibility diagram. b. Using the diagram, find the probability that the difference of the two numbers is (i) 1, (ii) non-zero, (iii) odd, (iv) a prime number, (v) more than 2. 35