Unit 6: What Do You Expect? Investigation 2: Experimental and Theoretical Probability
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1 Unit 6: What Do You Expect? Investigation 2: Experimental and Theoretical Probability Lesson Practice Problems Lesson 1: Predicting to Win (Finding Theoretical Probabilities) 1-3 Lesson 2: Choosing Marbles (Developing Probability Models) 5-7 Lesson 3: Designing a Fair Game (Pondering Possible and Probable) 8-11 Lesson 4: Winning the Bonus Prize (Using Strategies to Find Theoretical Probabilities) In the last Investigation, you collected the results of many coin tosses. You found that the experimental probability of a coin landing on heads is ½ (or very close to ½). You assume that the coins are fair coins for which there are two equally likely results of a toss, heads or tails. The word outcome means. The coin-tossing experiment had two possible outcomes, heads and tails. Heads was a favorable outcome for Kalvin. A probability calculated by examining possible outcomes, rather than by experimenting, is a. The probability of tossing heads is, or. The probability of tossing tails is also. But all experiments do not result in equally likely outcomes. When you tossed a cup, the two outcomes were not equally likely. The chances of landing on the side and landing in an upright position were not the same. In this investigation, you will explore some other situations in which probabilities are found both by experimenting and by analyzing the possible outcomes.
2 Lesson 1 Predicting to Win: Finding Theoretical Probabilities In the last 5 minutes of the Gee Whiz Everyone Wins! game show, all the members of the audience are called to the stage. Each one chooses a block at random from a bucket containing an unknown number of red, yellow, and blue blocks. Each block has the same size and shape. Before choosing, each contestant predicts the color of his or her block. If the prediction is correct, the contestant wins. After each selection, the block is put back into the bucket and the bucket is shaken. That way, the probabilities do not change as blocks are removed. What does random mean? Suppose you are a member of the audience. Would you rather be called to the stage first or last? Explain. Does it matter that the block is returned to the bucket and the bucket is shaken after each contestant? Explain. Problem 2.1 A. Play the block-guessing game with your class. Keep a record of the number of times a color is chosen. Play the game until you think you can predict the chances of each color being chosen. Red- Yellow- Blue-
3 1. Based on the data you collect during the game, find the experimental probabilities of choosing red, choosing yellow, and choosing blue. Red- Yellow- Blue- B. Suppose you counted the red blocks, the blue blocks, and the yellow blocks in the bucket. How would you use this information to calculate the theoretical probability of drawing a red, a blue, or yellow block. 1. How do the theoretical probabilities compare to the experimental probabilities in Question A? 2. What is the sum of the theoretical probabilities in Questions B? C. Does each individual block, without regard to color, have the same chance of being chosen? Explain. 1. Does each color have the same chance of being chosen? Explain. 2. Which person has the advantage- the first person to choose from the bucket or the last person? Explain.
4 Lesson 2 Choosing Marbles: Developing Probability Models Sammy collects marbles. He asks his teacher if the class could experiment with marbles instead of blocks. The teacher says, What really matters is whether we can predict the probabilities in a situation using marbles. Let s try a bag with marbles of different colors. Problem 2.2 A. A bag contains two yellow marbles, four blue marbles, and six red marbles. You choose a marble form the bag at random. Answer the following questions and explain your reasoning. 1. What is the probability the marbles is yellow? The probability it is blue? The probability it is red? 2. What is the sum of the probabilities from part (1)? 3. What color is the selected marble most likely to be? 4. What is the probability the marble is not blue? 5. What is the probability the marble is either red or yellow?
5 6. What is the probability the marble is white? B. Suppose a new bag has twice as many marbles of each color. 1. Do the probabilities change? Explain. 2. How many blue marbles should you add to this bag to have the probability of choosing a blue marble equal to? C. A different bag contains several marbles. Each marble is red or white or blue. The probably of choosing a red marble is, and the probability of choosing a white marble is. 1. What is the probability of choosing a blue marble? Explain. 2. What is the least number of marbles that can be in the bag? Suppose the bag contains the least number of marbles. How many of each color does the bag contain? 3. Can the bag contain 48 marbles? If so, how many of each color does it contain? 4. Suppose the bag contains 8 red marbles and 4 white marbles. How many blue marbles does it contain?
6 Lesson 3 Designing a Fair Game: Pondering Possible and Probable Santo and Tevy are playing a game with coins. They take turns tossing three coins. If all three coins match, Santo Wins. Otherwise, Tevy wins. Each player has won several turns in the game. Tevy, however, seems to be winning more often. Santo thinks the game is unfair. What is a Fair Game? Santo drew the tree diagram below to represent tossing three coins. What is a tree diagram? What is a sample space?
7 Problem 2.3 A. Use the tree diagram to answer the following questions. 1. What is the sample space for tossing three coins? 2. How many possible outcomes are there when you toss three coins? Are the outcomes equally likely? 3. What is the theoretical probability that the tree coins will match? 4. What is the theoretical probability that exactly two coins will match? 5. Is the game played by Santo and Tevy a fair game? If so, explain why. If not, explain how to make it fair. B. Suppose you toss three coins for 24 trials. How many times would you expect two coins to match? C. Santo said, It is possible to toss three coins and have them match. Tevy replied, Yes, but is it probable? What do you think each boy meant?
8 Lesson 4 Winning the Bonus Prize: Using Strategies to Find Theoretical probabilities Allt he winners from the Gee Whiz Everyone Wins! game show have the opportunity to compete for a bonus prize. Each winner chooses one block from each of two bags. Each bag contains one red, one yellow, and one blue block. This bonus game consists of two events, which can also be called a. The contestant must predict which color she or he will choose from each of the two bags. If the prediction is correct, the contestant wins a $10,000 bonus prize! Problem 2.4 A. Conduct an experiment with 36 trials for the situation above. Record the pairs of colors that you choose. Trial Result
9 Find the experimental probability of choosing each possible pair of colors. B. Find all the possible color pairs that can be chosen. Are these outcomes equally likely? Explain your reasoning. 1. Find the theoretical probability of choosing each pair of colors. 2. How do the theoretical probabilities compare with your experimental probabilities? Explain any differences.
Date Learning Target/s Classwork Homework Self-Assess Your Learning. Pg. 2-3: WDYE 2.3: Designing a Fair Game
What Do You Expect: Probability and Expected Value Name: Per: Investigation 2: Experimental and Theoretical Probability Date Learning Target/s Classwork Homework Self-Assess Your Learning Mon, Feb. 29
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