11.2 Probability of Compound Events Lesson Objective • Use possibility diagrams to find probability of compound events. Use Possibility Diagrams to Find Probability of Compound Events. You have seen how to find all possible outcomes of a compound event by drawing a possibility diagram. From a possibility diagram, you are able to count the number of favorable outcomes and the total number of outcomes. You can then find the probability of a compound event E using the definition of probability: P(E ) 5 Number of outcomes favorable to event E Total number of equally likely outcomes You have learned how to draw a tree diagram to represent the possible outcomes of tossing coins. Consider finding the probability of landing heads, then tails when tossing a coin twice. 1st Toss 2nd Toss Outcome H (H, H) T (H, T) H (T, H) H T T (T, T) H represents heads T represents tails You can see that there are four possible outcomes: (H, H), (H, T), (T, H), and (T, T). However, there is only one favorable outcome, (H, T). So, P(H, T) 5 1 4 You want to find the probability of landing heads then tails, so the order of events is important. P(H, T) is not the same as P(T, H) here. Lesson 11.2 Probability of Compound Events (M)MIFSEC3B_Ch11.indd 229 229 10/18/12 6:44 PM Example 4 Use a possibility diagram to find the probability of a compound event. Use a possibility diagram to find each probability. a) A fair coin and a fair six-sided number die are tossed together. Find the probability of showing heads and a 5. Solution Method 1 T Coin Tossed H 1 2 3 4 5 H represents heads T represents tails 6 Number Die Rolled P(H, 5) 5 1 12 Think Math Would the probability be different if the possibility diagram were drawn with the axis labels interchanged? Explain your answer. You can see that there is only one outcome with heads and a 5. Method 2 Coin Number Die 1 2 3 4 5 6 H (1, H) (2, H) (3, H) (4, H) (5, H) (6, H) T (1, T) (2, T) (3, T) (4, T ) (5, T ) (6, T ) H represents heads T represents tails P(H, 5) 5 P(5, H) 1 5 12 230 Chapter 11 Probability (M)MIFSEC3B_Ch11.indd 230 10/18/12 6:44 PM b) Two fair six-sided number dice are rolled. Find the probability that the sum of the two numbers rolled is a prime number. Solution 2nd Number Die 1st Number Die Math Note + 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 Recall that a prime number is a whole number greater than 1 that is divisible by only 1 and itself. The prime numbers in the possibility diagram are 2, 3, 5, 7, and 11. There are 15 prime numbers out of 36 equally likely possible outcomes. P(sum is prime) 5 15 36 5 5 12 c) Two fair four-sided number dice, one red (R) and one blue (B), are rolled, and the number on the bottom is recorded. The red number die has numbers 1, 2, 4, and 7. The blue number die has numbers 2, 5, 8, and 9. Find the probability that the number recorded from the blue number die is more than 3 greater than the number recorded from the red number die. That is, find P(B 2 R 3). Solution Method 1 Blue Number Die Red Number Die 1 2 4 7 2 (1, 2) (2, 2) (4, 2) (7, 2) 5 (1, 5) (2, 5) (4, 5) (7, 5) 8 (1, 8) (2, 8) (4, 8) (7, 8) 9 (1, 9) (2, 9) (4, 9) (7, 9) P(B 2 R 3) 5 7 16 Continue on next page Lesson 11.2 Probability of Compound Events (M)MIFSEC3B_Ch11.indd 231 231 10/18/12 6:44 PM Method 2 You can list the differences instead of the numbers Blue Number Die Red Number Die rolled. Then look for 2 1 2 4 7 2 1 0 22 25 5 4 3 1 22 8 7 6 4 1 9 8 7 5 2 P(B R 3) 5 differences greater than 3. 7 16 d) The two spinners shown are spun. Find the probability that the pointers stop at 1 on Spinner 1 and blue (B) on Spinner 2. 1 1 2 4 B (1, B) (1, B) (2, B) (4, B) G (1, G) (1, G) (2, G) (4, G ) R (1, R) (1, R) (2, R) (4, R ) G 2 Spinner 1 Spinner 1 Spinner 2 1 B 4 Solution P(1, B) 5 1 R Spinner 2 2 1 5 12 6 Guided Practice Draw a possibility diagram to find each probability. 1 Two fair four-sided number dice, each numbered 1 to 4, are rolled together. The result recorded is the number facing down. Find the probability that the product of the two numbers is divisible by 2. 2 One colored disc is randomly drawn from each of two bags. Each bag has 5 colored discs: 1 red, 1 green, 1 blue, 1 yellow, and 1 white. Find the probability of drawing a blue or yellow disc. 3 A box has 1 black, 1 green, 1 red, and 1 yellow marble. Another box has 1 white, 232 1 green, and 1 red marble. A marble is taken at random from each box. Find the probability that a red marble is not drawn. Chapter 11 Probability (M)MIFSEC3B_Ch11.indd 232 10/18/12 6:44 PM Example 5 Use tree diagrams to find probabilities of compound events. Suppose that it is equally likely to rain or not rain on any given day. Draw a tree diagram and use it to find the probability that it rains exactly once on two consecutive days. 1st Day 2nd Day Outcome R (R, R) R9 (R, R9) R (R9, R) R R9 R9 (R9, R9) R represents raining R9 represents not raining The favorable outcome is “rain exactly once on two consecutive days.” So, you should look for the outcome that gives (R, R9) or (R9, R), meaning either it rains on the first day or it rains on the second day. Solution P(rain exactly once on two consecutive days) 5 2 4 5 1 2 1 2 So, the probability of (R, R′) or (R′, R) is . Guided Practice Solve. Show your work. 4 Three fair coins are tossed together. a) Draw a tree diagram to represent all possible outcomes. b) Using your answer in a), find the probability of getting all heads. c) Using your answer in a), find the probability of getting at least two tails. Lesson 11.2 Probability of Compound Events (M)MIFSEC3B_Ch11.indd 233 233 10/18/12 6:44 PM Practice 11.2 Solve. Show your work. 1 A bag contains 2 blue balls and 1 red ball. Winnie randomly draws a ball from the bag and replaces it before she draws a second ball. Use a possibility diagram to find the probability that the balls drawn are different colors. 2 A letter is randomly chosen from the word FOOD, followed by randomly choosing a letter from the word DOG. Draw a tree diagram to find the probability that both letters chosen are the same. 3 Three pebbles are placed in a bag: 1 blue, 1 green, and 1 yellow. First a pebble is randomly drawn from the bag. Then a fair four-sided number die labeled 1 to 4 is rolled. The result recorded is the number facing down. Use a possibility diagram to find the probability of drawing a yellow pebble and getting a 4. 4 Thomas rolled two number dice: a fair six-sided one and a fair four-sided one labeled 1 to 4. Use a possibility diagram to find the probability of rolling the number 3 on both dice. 5 A bike shop has 3 bikes with 20-speed gears and 2 bikes with 18-speed gears. The bike shop also sells 1 blue helmet and 1 yellow helmet. Use a possibility diagram to find the probability of getting an 18-speed bicycle and a blue helmet if one bike and one helmet is randomly selected from the shop. 234 Chapter 11 Probability (M)MIFSEC3B_Ch11.indd 234 10/18/12 6:44 PM 6 Susan randomly draws a card from three number cards: 1, 3, and 6. After replacing the card, Susan randomly draws another number card. The product of the two numbers drawn is recorded. 1 3 6 a) Draw a possibility diagram to represent all possible outcomes. b) Using your answer in a), what is the probability of forming a number greater than 10 but less than 30? 7 Jane and Jill watch television together for 2 hours. Jane selects the channel for the first hour, and Jill selects the channel for the second hour. Jane’s remote control randomly selects from Channels A, B, and C. Jill’s remote control randomly selects from Channels C, D, and E. a) Draw a possibility diagram to represent all possible outcomes for the channels they watch on television for 2 hours. b) Using your answer in a), what is the probability of watching the same channel for both hours? 8 A color disc is randomly drawn from a bag that contains the following discs. Y R B R G After a disc is drawn, a fair coin is tossed. Use a possibility diagram to find the probability of drawing a red disc and landing on heads. 9 Karen tosses a fair coin three times. Draw a tree diagram to find the probability of getting the same result in all three tosses. 10 Mrs. Bridget wants to make pancakes. She has 3 types of flour: cornmeal flour, whole wheat flour, and white flour. She also has 2 cartons of milk: whole milk and low fat milk. However, all the types of flour and milk are not labeled, so she randomly guesses which types of flour and milk to use for her pancakes. a) Draw a tree diagram to represent all possible outcomes. b) Using your answers in a), find the probability that she will use cornmeal flour and low fat milk for her pancakes. Lesson 11.2 Probability of Compound Events (M)MIFSEC3B_Ch11.indd 235 235 10/18/12 6:44 PM
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