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4/8/13 9:05 AM Name Page 2 Class Date Reteaching 9-1 Probability To find a theoretical probability, first list all possible outcomes. Then use the formula: number of favorable outcomes P(event) = total number of possible outcomes A letter is selected at random from the letters of the word FLORIDA. What is the probability that the letter is an A? Selecting a letter other than A is called not A and is the complement of the event A. The sum of the probabilities of an event and its complement equals 1, or 100%. What is the probability of the event “not A”? P(A) P(not A) 1 1 7 • There are 7 letters (possible outcomes). P(not A) 1 6 1 P(not A) 1 7 7 • There is one A, which represents a favorable outcome. All rights reserved. 000200010271713722_CH09_L01-L02.qxd The probability of the event “not A” of favorable outcomes 1 P(A) number total number of outcomes = 7 (selecting F, L, O, R, I, or D) is 67. The probability that the letter is an A is 17. Spin the spinner shown once. Find each probability as a fraction, a decimal, and a percent. 1. P(5) 5 2. P(odd number) number of favorable outcomes total number of outcomes 7 2 number of favorable outcomes total number of outcomes 8 10 = 5 © Pearson Education, Inc., publishing as Pearson Prentice Hall. 2 = You select a card at random from a box that contains cards numbered from 1 to 10. Find each probability as a fraction, a decimal, and a percent. 3. P(even number) 4. P(number less than 4) 5. P(not 5) The letters H, A, P, P, I, N, E, S, and S are written on pieces of paper. Select one piece of paper. Find each probability. 6. P(not vowel) 7. P(not E) A number is selected at random from the numbers 1 to 50. Find the odds in favor of each outcome. 8. selecting a multiple of 5 Course 2 Lesson 9-1 9. selecting a factor of 50 Reteaching 000200010271713722_CH09_L01-L02.qxd 4/8/13 9:06 AM Name Page 6 Class Date Reteaching 9-2 Experimental Probability Probability measures how likely it is that an event will occur. For an experimental probability, you collect data through observations or experiments and use the data to state the probability. G B G R R B G G B R B B R G R G B B G The jar contains red, green, and blue chips. You shake the jar, draw a chip, note its color, and then put it back. You do this 20 times with these results: 7 blue chips, 5 red chips, and 8 green chips. The experimental probability of drawing a green chip is P(green chip) = number of times “green chips” occur total number of trials G All rights reserved. 8 = 2 = 0.4 = 40% P(green chip) = 20 5 The probability of drawing a green chip is 25, or 0.4, or 40%. Sometimes a model, or simulation, is used to represent a situation. Then, the simulaton is used to find the experimental probability. For example, spinning this spinner can simulate the probability that 1 of 3 people is chosen for president of the student body. 1. What is the experimental probability of drawing a red chip? Write the probability as a fraction. P(red chip) = 20 2. What is the experimental probability of drawing a blue chip? Write the probability as a percent. = P(blue chip) = = Suppose you have a bag with 30 chips: 12 red, 8 white, and 10 blue. You shake the jar, draw a chip, note its color, and then put it back. You do this 30 times with these results: 10 blue chips, 12 red chips, and 8 white chips. Write each probability as fraction in simplest form. 3. P(red) 4. P(white) 5. P(blue) Describe a probability simulation for each situation. 6. You guess the answers on a true/false test with 20 questions. Course 2 Lesson 9-2 7. One student out of 6 is randomly chosen to be the homeroom representative. Reteaching © Pearson Education, Inc., publishing as Pearson Prentice Hall. Use the 20 draws above to complete each exercise. 000200010271713722_CH09_L03.qxd 2/14/13 12:56 AM Name Page 2 Class Date Reteaching 9-3 Sample Spaces The set of all possible outcomes of an experiment is called the sample space. You can use a tree diagram or a table to show the sample space for an experiment. The tree diagram below shows the sample space for spinning the spinner and tossing a coin. H 1 3 2 Evelyn and Kara are planning to go skating or to a movie. Afterward they want to go out for pizza, tacos, or cheeseburgers. How many possible choices do they have? T H 2 T H • There are two choices for an activity and three choices for food. T • First choices ⫻ Second choices 3 H T 2 There are 6 possible outcomes: 1H, 1T, 2H, 2T, 3H, 3T. What is the probability of spinning a 3 and tossing heads? There is one favorable outcome (3H) out of 6 possible outcomes. The probabilty is 16. ⫻ 3 =6 There are 6 possible choices. © Pearson Education, Inc., publishing as Pearson Prentice Hall. 1 Complete the tree diagram to show the sample space. 1. Roll a number cube and toss a coin. What is the probability of getting (4, Heads)? 1 2 3 4 5 6 Number of outcomes P(4, heads) = Use the counting principle to find the number of possible outcomes. 2. 4 kinds of yogurt and 8 toppings 3. 6 shirts and 9 pairs of slacks 4. 3 types of sandwiches and 3 flavors 5. 4 types of bread and 6 different of juice Course 2 Lesson 9-3 All rights reserved. You can use the counting principle to find the number of possible outcomes: If there are m ways of making one choice and n ways of making a second choice, then there are m ⫻ n ways of making the first choice followed by the second. sandwich spreads Reteaching 4/8/13 9:26 AM Page 2 Name Class Date Reteaching 9-4 If you toss a coin and roll a number cube, the events are independent. The outcome of one event does not affect the outcome of the second event. Find the probability of tossing a heads (H) and rolling an even number (E). Find P(H and E). H and E are independent. 1 Compound Events If the outcome of the first event affects the outcome of the second event, the events are dependent. A bag contains 3 blue and 3 red marbles. Draw a marble, then draw a second marble without replacing the first marble. Find the probability of drawing 2 blue marbles. 1 Find P(H): 3 blue 5 1 P(blue) = 6 marbles 2 s 1 P(H) = 12 head sides 5 2 2 2 Find P(E): Find P(blue after blue). 2 blue 5 2 P(blue after blue) = 5 marbles 5 1 P(E) = 36 evens faces 5 2 3 Find P(blue). P(H and E) = P(H) ⫻ P(E) = 12 3 12 5 14 3 All rights reserved. 000200010271713722_CH09_L04.qxd Find P(blue, then blue) P(blue, then blue) = P(blue) ⫻ P(blue after blue) In Exercises 1–3, you draw a marble at random from the bag of marbles shown. Then, you replace it and draw again. Find each probability. 1. P(blue, then red) 2. P(2 reds) 3. P(2 blues) © Pearson Education, Inc., publishing as Pearson Prentice Hall. 5 12 3 25 5 15 B R R R B R R R B B Next, you draw two marbles randomly without replacing the first marble. Find each probability. 4. P(blue, then red) 5. P(2 reds) 6. P(2 blues) You draw two letters randomly from a box containing the letters M, I, S, S, O, U, R, and I. 7. Suppose you do not replace the first letter before drawing the second. What is P(M, then I)? 8. Suppose you replace the first letter before drawing the second. What is P(M, then I)? Course 2 Lesson 9-4 Reteaching 000200010271713722_CH09_L05.qxd 4/8/13 8:03 AM Name Page 2 Class Reteaching 9-5 Step 1 Date Simulating Compound Events Choose a simulation tool. • You can use a number cube when there are 6 equally likely outcomes. • You can use a spinner with x number of equal spaces when there are x equally likely outcomes. • You can use a coin when there are 2 equally likely outcomes. • You can use random digits. Step 2 Decide which outcomes are favorable. Step 3 All rights reserved. • Choose what you need to land on, roll, or toss to get a favorable outcome. Describe a trial. • For each trial, use your simulation tool until you get a favorable outcome. • Record the number of times you use your tool to get your favorable outcome. Step 4 Perform 20 trials. Then estimate the probability. One-fourth of the students in the seventh grade have no siblings. Design a simulation for estimating the probability that you would need to ask at least two students before finding one with no siblings. 1. Choose a simulation tool. 2. Decide which outcomes are favorable. Course 2 Lesson 9-5 Reteaching © Pearson Education, Inc., publishing as Pearson Prentice Hall. • Make a table showing your 20 trials and their outcomes. • Use that to find the probability: the number of favorable outcomes over 20. 000200010271713722_CH09_L05.qxd 2/14/13 Name 2:21 AM Page 3 Class Date Reteaching 9-5 (continued) Simulating Compound Events 3. Describe a trial. 4. Perform 20 trials. Students Asked to Find One with No Siblings Frequency 2 5. Find the probability that you would need to ask at least two students before finding one with no siblings. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. 1 Reteaching Course 2 Lesson 9-5