Topic : ADDITION OF PROBABILITIES (MUTUALLY EXCLUSIVE EVENTS) TIME : 4 X 45 minutes

Transcription

1 Worksheet 6 th Topic : ADDITION OF PROBABILITIES (MUTUALLY EXCLUSIVE EVENTS) 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: To determine the probability of the mutually exclusive events (Addition of Probabilities) Addition of Probabilities (Mutually Exclusive Events) Two or more events are said to be mutually exclusive if the occurrence of any one of them excludes the occurrence of the others (no common outcomes). Thus if E 1 and E 2 are mutually exclusive events, then P(E 1 and E 2 ) = P(E 1 E 2 ) = 0. Suppose "E 1 or E 2 " denotes the event that "either E 1 or E 2 both occur", then (a) If E 1 and E 2 are not mutually exclusive events: We can also write: P(E 1 or E 2 ) = P(E 1 ) + P(E 2 ) P(E 1 and E 2 ) P(E 1 E 2 ) = P(E 1 ) + P(E 2 ) P(E 1 E 2 ) (b) If E 1 and E 2 are mutually exclusive events: P(E 1 or E 2 ) = P(E 1 ) + P(E 2 ) In general for not two mutually exclusive events A and B, P ( A occurs or B occurs) = 46

2 P( A or B) = P( A) + P( B) P( Aand B) In general for two mutually exclusive events A and B, P ( A occurs or B occurs) = P( A or B) = P( A) + P( B) The result can be extended to cover more than 2 mutually exclusive events, P ( A occurs or B occurs or C occurs) = P( A or B or C ) = Example 43 Box A contains 4 pieces of paper numbered 1, 2, 3, 4. Box B contains 2 pieces of paper numbered 1, 2. One piece of paper is removed at random from each box. Draw a tree diagram to display all the possible outcomes of the experiment. a. Using the diagram, find the probability of obtaining at least one 1 b. By adding the possible sums and product of the two numbers obtained for the diagram, find the probability that (i) the sum of two numbers is 3, (ii) the sum of two numbers is 5, (iii) the sum of two numbers is odd, (iv) the product of two numbers is exactly divisible by 4, (v) the product of two numbers is not a prime, (vi) the product of two numbers is at least 4, (vii) the sum is equal to the product. a. The tree diagram: From the diagram, n (S) = 47

3 a. Let A denote the event at least one 1, A = { } and n (A) = P (A) = b. We define the following events: B: the sum of two numbers is 3, C: the sum of two numbers is 5, D: the sum of two numbers is odd, E: the product of two numbers is exactly divisible by 4, F: the product of two numbers is not a prime, G: the product of two numbers is at least 4, H: the sum is equal to the product. (i) P (B) = (ii) P (C) = (iii) P (D) = (iv) (v) P (F) = (vi) (vii) P (H ) = P (E) P (G) = = Example 44 The probability of three teams L, M, and N, winning a football competition are respectively. Calculate the probability that a. either L or M wins b. neither L nor N wins a. P(L or M wins) = P(L wins) + P(M wins) = + = b. P(L or N wins) = P(..) + P(..) = + = P(neither L nor N wins) =. 1 1, 4 8, and 10 1 Example 45 A card is drawn at random from an ordinary pack of 52 playing cards. Find the probability that the card is a. an ace or king b. a heart or a diamond c. neither a king nor a queen We define the following events: A: the card drawn is an ace H: the card drawn is a heart 48

4 : the card drawn is a king D: the card drawn is a diamond Q: the card drawn is a queen a. P(A) =. and P() =.. P(A or ) = +... = =.. b. P(H) =. and P(D) =.. P(H or D) = +... = =.. c. P() =. and P(Q) =.. P( or Q) = +... = =.. P(neither nor Q) = 1.. = Example 46 It is known that the probability of obtaining zero defectives in a sample of 40 items is 0.34 whilst the probability of obtaining 1 defective item in the sample is What is the probability (a) obtaining not more than 1 defective item in a sample? (b) obtaining more than 1 defective items in a sample? Example 47 The probability that a student passes Mathematics is 3 2 and the probability that he passes English is 9 4. If the probability that he will pass at least one subject is 5 4, what is the probability that he will pass both subjects? (We assume it is based on probability only.) 49

5 Example 48 A box contains 100 items of which 4 are defective. Two items are chosen at random from the box. What is the probability of selecting (a) 2 defectives if the first item is not replaced; (b) 2 defectives if the first item is put back before choosing the second item; (c) 1 defective and 1 non-defective if the first item is not replaced? Example 49 Five small radios are packed in identical, unmarked individual sealed boxes. Three boxes are on table X and contain 2 radios made by firm A and one by firm B. Two boxes are on table Y and contain one radio made by firm A and one by firm B. If someone moves a box from table X to table Y and you randomly select a box from table Y, what is the probability that you will select a radio made by firm B? 50

6 Example 50 If the independent probabilities that three people A, B and C will be alive in 30 years time are 0.4, 0.3, 0.2 respectively, calculate the probability that in 30 years' time, (a) all will be alive (b) none will be alive (c) only one will be alive (d) at least one will be alive Exercise 9 1. A die in the form of a tetrahedron (solid regular triangular pyramid) has the numbers 1, 2, 3, and 4 printed on its four faces. a. When the die is thrown what is the probability that (i) it will land with the face printed 4 down, (ii) it will land so that the sum of the three upper faces is an odd number. b. If the die and a coin are thrown together, list all possible outcomes of the experiment using a tree diagram. 2. A spinner as shown below and a coin are used in a game. The spinner is spun once and the coin is tossed once. Draw a tree diagram to list all possible outcomes. With the help of the tree diagram, calculate the probability of getting a. red on the spinner and tail on the coin, Red b. blue or yellow on the spinner and head on the coin Blue Yellow 51

7 3. Two spinners similar to the spinner in Question 2 are spun simultaneously. Display all possible outcomes in a tree diagram. Hence, find the probability that a. the pointers will stop on sectors of different colours, b. one of the pointers stops on red and the other pointer stops on yellow, c. the first pointer stops on blue and the second pointer stops on red or yellow. 4. A bag contains 4 card numbers 1, 3, 5, 7. A second bag contains 3 cards numbered 1, 2, and 7. One card is drawn at random from each bag. a. Draw a tree diagram for the experiment. b. With the help of your tree diagram, calculate the probability that the two numbers obtained (i) have the same value, (ii) are both odd, (iii) are both prime, (iv) have a sum greater than 4, (v) have a sum that is even, (vi) have a product that is prime, (vii) have a product that is greater than 20, (viii) have a product that is divisible by Eleven cards numbered 11, 12, 13, 14,, 21 are placed in a box. A card is removed at random from the box. Find the probability that the number on the card is a. even, b. prime, c. either even or prime, d. divisible by 3, e. either even divisible by 3, f. odd, g. divisible by 4, h. either odd or divisible by A bag contains 7 red, 5 green, and 3 blue counters. A counter is selected at random from the bag. Find the probability of selecting a. a red counter, b. a green counter, c. either a red or a green counter, d. neither a red nor a green counter. 7. The letters of the word MUTUALLY and the word EXCLUSIVE are written on individual cards and the cards are put into a box. A card is picked at random. What is the probability of picking a. the letter U, b. the letter E, c. the letter U or E, d. a consonant, e. the letter U or a consonant, f. the letter U or E or L? 8. A coin is tossed three times. Display all the possible outcomes of the experiment using a tree diagram. From your tree diagram, find the probability of obtaining 52

8 a. three heads, b. exactly two heads, c. at least two heads. 9. The probability of football team winning any match is 10 7 and the probability of losing any match is 152. a. What is the probability that the team wins or loses a particular match? b. What is the probability that the team neither wins nor loses a match? 10. When a golfer plays any hole, the probabilities that he will take 4, 5, or 6 strokes are 141, 7 2, 7 3 respectively. He never takes less than 4 strokes. Calculate the probability that in playing a hole, he will take a. 4 or 5 strokes, b. 4, 5, or 6 strokes, c. more than 6 strokes. 11. In a basketball tournament, three of the participating teams, Panda, Spaceship, and Rocket have the probabilities 154, 101, and 5 1 respectively, of winning the tournament. Find the probability that a. Panda or Rocket will win the tournament, b. Panda, Spaceship, or Rocket will win the tournament, c. neither Panda nor Rocket will win the tournament, d. none of these three teams will win the tournament. 53