Probabilities of Compound Events

Transcription

1 0 LESSON Probabilities of Compound Events UNDERSTAND Sometimes, you may want to find the probability that two or more events will occur at the same time. This is called finding the probability of a compound event. Organized lists, diagrams, and tables can help you do this. Imagine tossing two fair coins. The outcome of the first event does not affect the outcome of the second event. So, these are called independent events. Imagine drawing a marble from a bag and, without replacing it, drawing a second marble from the bag. In this case, the outcome of the second event depends on the outcome of the first event. These are called dependent events. For an experiment, Chad will toss a fair coin and a number cube with faces numbered to 6 at the same time. What is the probability that the coin lands tails up and the cube comes up? What are the possible outcomes for each event? The possible outcomes for tossing a coin are: heads and tails. The possible outcomes for tossing a number cube are:,,, 4, 5, and 6. The outcome of tossing the coin does not affect the outcome of tossing the cube. They are independent events. Make a table to determine all the possible outcomes. Tossing Number Cube Tossing Coin Heads H- H- H- H-4 H-5 H-6 Tails T- T- T- T-4 T-5 T-6 The table shows that there is one favorable outcome, T-, out of possible outcomes. P 5 number of favorable outcomes number of possible outcomes 5 The probability of the coin landing tails up and the cube coming up is 70 Domain 5: Statistics and Probability

2 Connect Each spinner below is divided into equal-sized sections. What is the probability that the first spinner will land on a shaded section and the second will land on the letter X? X Z What are the possible outcomes for each spinner? The possible outcomes for the first spinner are: shaded and unshaded. The possible outcomes for the second spinner are: X,, and Z. Make an organized list showing all the possible outcomes for spinning both spinners. shaded- X unshaded- X What is the probability that the first spinner will land on a shaded section and the second will land on the letter X? shaded- shaded- Z unshaded- unshaded- Z There is only favorable outcome, shaded-x. There are 6 possible outcomes. The probability of the first spinner landing on a shaded section and the second landing on the letter X is 6 TR What would be the probability of the first spinner landing on a shaded section and the second spinner landing on the letter W? Explain. Lesson 0: Probabilities of Compound Events 7

3 EXAMPLE A Maya tosses a quarter and a dime into the air at the same time. Make a tree diagram to show all possible outcomes. Then determine the probability that both coins will land on tails. Draw branches to show the outcomes of tossing the quarter. The possible outcomes of tossing the quarter are heads (H) and tails (T). Quarter H T From the ends of the branches you already drew, draw additional branches to show the possible outcomes of tossing the dime. The possible outcomes of tossing the dime are heads (H) and tails (T). Draw 4 additional branches to show the outcomes of tossing the quarter and dime together. Determine the probability that both coins will land tails up. There is possible outcome in which both coins land tails up, TT. Quarter Dime Outcomes H HH H T HT H TH T T TT There are a total of 4 possible outcomes. The tree diagram in step shows the entire sample space. The probability of both coins landing tails up is 4 CHECK Check your answer by using a different method to find all the possible outcomes and the probability of both coins landing tails up, such as making a table or an organized list. Show your work. 7 Domain 5: Statistics and Probability

4 EXAMPLE A bag contains blue () and green () marbles. Harrison will reach into the bag and pick a marble without looking. Without replacing the first marble, he will pick a second marble without looking. What is the probability that he will pick a green marble first and a blue marble second? Use branches to show the possible outcomes of Harrison s first pick. Draw additional branches to show the possible outcomes of his second pick. First Pick Since the first marble will not be replaced, the events are dependent. If he picks a blue marble first, then he must pick a green marble second. If he picks a green marble first, then he can pick either a blue marble or a green marble second. First Pick Second Pick Outcomes Determine the probability that he will pick a green marble first and a blue marble second. There are favorable outcomes,. There are 6 possible outcomes. The probability that Harrison will pick a green marble first and a blue marble second is 6, or DISCUSS Suppose Harrison will replace the first marble before picking the second marble. How would that change the probability described above? Lesson 0: Probabilities of Compound Events 7

5 Practice Identify each pair of events as independent or dependent.. drawing a marble from a bag, replacing it, and drawing a second marble. drawing a marble from a bag, not replacing it, and drawing a second marble HINT Does the outcome of the first event affect the second event? For each pair of spinners, determine the probability of spinning the first spinner so it lands on and spinning the second spinner so it lands on. Assume each spinner is divided into equal-sized sections. Show your work.. 4. A C A D C Two number cubes, with faces numbered to 6, are rolled. The table on the right can be used to represent all the possible outcomes of this experiment. Use the table for questions 5 7. Simplify if possible. 5. The outcome (, ) has been recorded for you in the table. This outcome shows rolling a on each cube. Use ordered pairs to represent each outcome in the table. 6. What is the probability of rolling double 6s? (, ) What is the probability of rolling doubles (two s, two s, etc.)? 74 Domain 5: Statistics and Probability

6 In the tree diagram below on the right, R stands for a red tile and stands for a yellow tile. Use the tree diagram for questions 8 and The tree diagram represents the possible outcomes in an experiment in which en draws a tile from a bag without looking and records its color. He then draws a second tile from the bag and records its color. ased on the tree diagram, does en replace the tiles after the first draw, or not? Explain your reasoning. en s Experiment First Pick R Second Pick R R 9. What is the probability that two tiles of the same color will be drawn in this experiment? Explain. Solve. 0. SHOW elow is a number cube, with faces numbered to 6, and a spinner divided into three congruent sections. Cassie will toss the cube and spin the spinner. What is the probability that her toss will result in a and the spinner will land on a shaded section? Show your work CREATE The bag shown has green () and blue () marbles in it. Jayson will choose a marble from this bag, and without replacing it, he will choose a second marble from the bag. On a separate sheet of paper, create a tree diagram to show all the possible outcomes of this experiment. Then determine the probability that both marbles chosen will be blue. Lesson 0: Probabilities of Compound Events 75