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10-4 Theoretical Probability Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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Warm Up 1. If you roll a number cube, what are the possible outcomes? 2. Add Add +. 1, 2, 3, 4, 5, or 6 Course Theoretical Probability

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Problem of the Day A spinner is divided into 4 different-colored sections. It is designed so that the probability of spinning red is twice the probability of spinning green, the probability of spinning blue is 3 times the probability of spinning green, and the probability of spinning yellow is 4 times the probability of spinning green. What is the probability of spinning yellow? Course Theoretical Probability 0.4

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Learn to estimate probability using theoretical methods. Course Theoretical Probability

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Vocabulary theoretical probability equally likely fair mutually exclusive disjoint events Insert Lesson Title Here Course Theoretical Probability

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Course Theoretical Probability Theoretical probability is used to estimate probabilities by making certain assumptions about an experiment. Suppose a sample space has 5 outcomes that are equally likely, that is, they all have the same probability, x. The probabilities must add to 1. x + x + x + x + x = 1 5x = 1 x = 1 5

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Course Theoretical Probability A coin, die, or other object is called fair if all outcomes are equally likely.

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An experiment consists of spinning this spinner once. Find the probability of each event. Example 1A: Calculating Theoretical Probability Course Theoretical Probability P(4) The spinner is fair, so all 5 outcomes are equally likely: 1, 2, 3, 4, and P(4) = = number of outcomes for 4 5

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Check It Out: Example 1A Course Theoretical Probability P(1) An experiment consists of spinning this spinner once. Find the probability of each event. The spinner is fair, so all 5 outcomes are equally likely: 1, 2, 3, 4, and P(1) = = number of outcomes for 1 5

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Example 1B: Calculating Theoretical Probability Course Theoretical Probability P(even number) There are 2 outcomes in the event of spinning an even number: 2 and 4. P(even number) = number of possible even numbers = An experiment consists of spinning this spinner once. Find the probability of each event.

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Check It Out: Example 1B Course Theoretical Probability P(odd number) There are 3 outcomes in the event of spinning an odd number: 1, 3, and 5. P(odd number) = number of possible odd numbers = An experiment consists of spinning this spinner once. Find the probability of each event.

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An experiment consists of rolling one fair number cube and flipping a coin. Find the probability of the event. Example 2A: Calculating Probability for a Fair Number Cube and a Fair Coin Course Theoretical Probability Show a sample space that has all outcomes equally likely. The outcome of rolling a 5 and flipping heads can be written as the ordered pair (5, H). There are 12 possible outcomes in the sample space. 1H2H3H4H5H6H 1T2T3T4T5T6T

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Check It Out: Example 2A Course Theoretical Probability P(one head & one tail) There are 2 outcomes in the event getting one head and getting one tail: (H, T) and (T, H). P(head and tail) = = An experiment consists of flipping two coins. Find the probability of each event.

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Example 2B: Calculating Theoretical Probability for a Fair Coin Course Theoretical Probability P(any number and tails) There are 6 outcomes in the event flipping tails: (1, T), (2, T), (3, T), (4, T), (5, T), and (6, T). P(tails) = = An experiment consists of flipping a coin and rolling a number cube at the same time. Find the probability of the event.

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Check It Out: Example 2B Course Theoretical Probability P(both tails) There is 1 outcome in the event both tails: (T, T). P(both tails) = 1 4 An experiment consists of flipping two coins. Find the probability of each event.

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Carl has 3 green buttons and 4 purple buttons. How many white buttons should be added so that the probability of drawing a purple button is ? Example 3: Calculating Theoretical Probability Course Theoretical Probability Adding buttons to the bag will increase the number of possible outcomes. Let x equal the number of white buttons. Set up a proportion x = 2 9 2(7 + x) = 9(4) Find the cross products. 2 9

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Example 3: Calculating Theoretical Probability Continued Course Theoretical Probability Multiply x = 36 –14 – 14Subtract 14 from both sides. 2x = 22 x = 11 Divide both sides by white buttons should be added to the bag. 2 2

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Stephany has 2 dimes and 3 nickels. How many pennies should be added so that the probability of drawing a nickel is ? Check It Out: Example 3 Course Theoretical Probability Adding pennies to the bag will increase the number of possible outcomes. Let x equal the number of pennies. Set up a proportion x = 3 7 3(5 + x) = 3(7) Find the cross products. 3 7

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Check It Out: Example 3 Continued Course Theoretical Probability Multiply x = 21 –15 – 15 Subtract 15 from both sides. 3x = 6 x = 2 Divide both sides by 3. 2 pennies should be added to the bag. 3

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Course Theoretical Probability Two events are mutually exclusive, or disjoint events, if they cannot both occur in the same trial of an experiment. For example, rolling a 5 and an even number on a number cube are mutually exclusive events because they cannot both happen at the same time. Suppose both A and B are two mutually exclusive events. P(both A and B will occur) = 0 P(either A or B will occur) = P(A) + P(B)

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Example 4: Find the Probability of Mutually Exclusive Events Course Theoretical Probability Suppose you are playing a game in which you roll two fair number cubes. If you roll a total of five you will win. If you roll a total of two, you will lose. If you roll anything else, the game continues. What is the probability that you will lose on your next roll? P(game ends) = P(total = 2) The event total = 2 consists of 1 outcome, (1, 1), so P(total = 2) = The probability that you will lose is, or about 3% =

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Check It Out: Example 4 Course Theoretical Probability Suppose you are playing a game in which you flip two coins. If you flip both heads you win and if you flip both tails you lose. If you flip anything else, the game continues. What is the probability that the game will end on your next flip? It is impossible to flip both heads and tails at the same time, so the events are mutually exclusive. Add the probabilities to find the probability of the game ending on your next flip.

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Check It Out: Example 4 Continued Course Theoretical Probability P(game ends) = P(both tails) + P(both heads) The event both heads consists of 1 outcome, (H, H), so P(both heads) =. The event both tails consists of 1 outcome, (T, T), so P(both tails) = = = 1 2 The probability that the game will end is, or 50%. 1 2

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Lesson Quiz An experiment consists of rolling a fair number cube. Find each probability. 1. P(rolling an odd number) 2. P(rolling a prime number) An experiment consists of rolling two fair number cubes. Find each probability. 3. P(rolling two 3s) 4. P(total shown > 10) Insert Lesson Title Here Course Theoretical Probability

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