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**Theoretical Probability**

9-4 Theoretical Probability Warm Up Problem of the Day Lesson Presentation Pre-Algebra

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**Theoretical Probability**

Pre-Algebra 9-4 Theoretical Probability Warm Up 1. If you roll a number cube, what are the possible outcomes? 2. Add 3. Add 1, 2, 3, 4, 5, or 6 3 36 1 6 1 4 1 2 2 36 5 9

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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? 0.4

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**Learn to estimate probability using theoretical methods.**

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**Vocabulary theoretical probability equally likely fair**

mutually exclusive

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

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**Additional Example 1A: Calculating Theoretical Probability**

An experiment consists of spinning this spinner once. A. What is the probability of spinning a 4? The spinner is fair, so all 5 outcomes are equally likely. The probability of spinning a 4 is P(4) = . 1 5

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**Additional Example 1B: Calculating Theoretical Probability**

An experiment consists of spinning this spinner once. B. What is the probability of spinning an even number? There are 2 outcomes in the event of spinning an even number: 2 and 4. P(spinning an even number) = number of possible even numbers 5 2 5 =

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**Additional Example 1C: Calculating Theoretical Probability**

An experiment consists of spinning this spinner once. C. What is the probability of spinning a number less than 4? There are 3 outcomes in the event of spinning a number less than 4: 1, 2, and 3. 3 5 P(spinning a number less than 4)=

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Try This: Example 1A An experiment consists of spinning this spinner once. A. What is the probability of spinning a 1? The spinner is fair, so all 5 outcomes are equally likely. The probability of spinning a 1 is P(1) = . 1 5

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**An experiment consists of spinning this spinner once.**

Try This: Example 1B An experiment consists of spinning this spinner once. B. What is the probability of spinning an odd number? There are 3 outcomes in the event of spinning an odd number: 1, 3, and 5. P(spinning an odd number) = number of possible odd numbers 5 3 5 =

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Try This: Example 1C An experiment consists of spinning this spinner once. C. What is the probability of spinning a number less than 3? There are 2 outcomes in the event of spinning a number less than 3: 1 and 2. 2 5 P(spinning a number less than 3)=

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**Additional Example 2A: Calculating Theoretical Probability for a Fair Die and a Fair Coin**

An experiment consists of rolling one fair die and flipping a coin. A. 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. 1H 2H 3H 4H 5H 6H 1T 2T 3T 4T 5T 6T

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**Additional Example 2B: Calculating Theoretical Probability for a Fair Die and a Fair Coin**

An experiment consists of rolling one fair die and flipping a coin. B. What is the probability of getting 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) = = 6 12 1 2

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**Additional Example 2C: Calculating Theoretical Probability for a Fair Die and a Fair Coin**

An experiment consists of rolling one fair die and flipping a coin. C. What is the probability of getting an even number and heads? There are 3 outcomes in the event “getting an even number and heads”: (2, H), (4, H), (6, H). P(even number and heads) = = 3 12 1 4

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**Additional Example 2D: Calculating Theoretical Probability for a Fair Die and a Fair Coin**

D. What is the probability of getting a prime number? There are 6 outcomes in the event “getting a prime number”: (2, T), (3, T), (5, T), (2, H), (3, H), (5, H). P(prime number) = = 6 12 1 2

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Try This: Example 2A An experiment consists of flipping two coins. A. Show a sample space that has all outcomes equally likely. The outcome of flipping two heads can be written as HH. There are 4 possible outcomes in the sample space. HH TH HT TT

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**Try This: Example 2 Continued**

An experiment consists of flipping two coins. B. What is the probability of getting one head and 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) = = 2 4 1

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Try This: Example 2C An experiment consists of flipping two coins. C. What is the probability of getting heads on both coins? There is 1 outcome in the event “both heads”: (H, H). P(both heads) = 1 4

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Try This: Example 2D An experiment consists of flipping two coins. D. What is the probability of getting both tails? There is 1 outcome in the event “both tails”: (T, T). P(both tails) = 1 4

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Two events are mutually exclusive if they cannot both occur in the same trial of an experiment. 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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**Additional Example 3: Find the Probability of Mutually Exclusive Events**

Suppose you are playing a game in which you roll two fair dice. 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 the game will end on your next roll? It is impossible to roll a total of 5 and a total of 2 at the same time, so the events are mutually exclusive. Add the probabilities to find the probability of the game ending on your next roll.

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**Additional Example 3 Continued**

The event “total = 5” consists of 4 outcomes, (1, 4), (2, 3), (3, 2), and (4, 1), so P(total = 5) = . The event “total = 2” consists of 1 outcome, (1, 1), so P(total = 2) = . 1 36 4 P(game ends) = P(total = 5) + P(total = 2) 4 36 1 = + 5 36 = The probability that the game will end is , or about 13.9%. 5 36

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Try This: Example 3 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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**Try This: Example 3 Continued**

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 4 P(game ends) = P(both tails) + P(both heads) = + 1 4 = 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 die. Find each probability. 1. P(rolling an odd number) 2. P(rolling a prime number) An experiment consists of rolling two fair dice. Find each probability. 3. P(rolling two 3’s) 4. P(total shown > 10)

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