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**Theoretical Probability**

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

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**Theoretical Probability**

Course 3 10-4 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 1 12 1 6 1 4 1 2 2 36 5 9

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**Theoretical Probability**

Course 3 10-4 Theoretical Probability 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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**Theoretical Probability**

Course 3 10-4 Theoretical Probability Learn to estimate probability using theoretical methods.

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**Insert Lesson Title Here**

Course 3 10-4 Theoretical Probability Insert Lesson Title Here Vocabulary theoretical probability equally likely fair mutually exclusive disjoint events

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**Theoretical Probability**

Course 3 10-4 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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**Theoretical Probability**

Course 3 10-4 Theoretical Probability A coin, die, or other object is called fair if all outcomes are equally likely.

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**Example 1A: Calculating Theoretical Probability**

Course 3 10-4 Theoretical Probability Example 1A: Calculating Theoretical Probability An experiment consists of spinning this spinner once. Find the probability of each event. P(4) The spinner is fair, so all 5 outcomes are equally likely: 1, 2, 3, 4, and 5. 1 5 P(4) = = number of outcomes for

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**Theoretical Probability**

Course 3 10-4 Theoretical Probability Check It Out: Example 1A An experiment consists of spinning this spinner once. Find the probability of each event. P(1) The spinner is fair, so all 5 outcomes are equally likely: 1, 2, 3, 4, and 5. 1 5 P(1) = = number of outcomes for

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**Example 1B: Calculating Theoretical Probability**

Course 3 10-4 Theoretical Probability Example 1B: Calculating Theoretical Probability An experiment consists of spinning this spinner once. Find the probability of each event. 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 5 2 5 =

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**Theoretical Probability**

Course 3 10-4 Theoretical Probability Check It Out: Example 1B An experiment consists of spinning this spinner once. Find the probability of each event. 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 5 3 5 =

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**Theoretical Probability**

Course 3 10-4 Theoretical Probability Example 2A: Calculating Probability for a Fair Number Cube and a Fair Coin An experiment consists of rolling one fair number cube and flipping a coin. Find the probability of the event. 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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**Theoretical Probability**

Course 3 10-4 Theoretical Probability Check It Out: Example 2A An experiment consists of flipping two coins. Find the probability of each event. 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) = = 2 4 1

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

Course 3 10-4 Theoretical Probability Example 2B: Calculating Theoretical Probability for a Fair Coin An experiment consists of flipping a coin and rolling a number cube at the same time. Find the probability of the event. 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) = = 6 12 1 2

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**Theoretical Probability**

Course 3 10-4 Theoretical Probability Check It Out: Example 2B An experiment consists of flipping two coins. Find the probability of each event. P(both tails) There is 1 outcome in the event “both tails”: (T, T). P(both tails) = 1 4

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**Example 3: Calculating Theoretical Probability**

Course 3 10-4 Theoretical Probability Example 3: Calculating Theoretical Probability 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 ? 2 9 Adding buttons to the bag will increase the number of possible outcomes. Let x equal the number of white buttons. 4 7 + x = 2 9 Set up a proportion. 2(7 + x) = 9(4) Find the cross products.

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**Example 3: Calculating Theoretical Probability Continued**

Course 3 10-4 Theoretical Probability Example 3: Calculating Theoretical Probability Continued 14 + 2x = 36 Multiply. – – 14 Subtract 14 from both sides. 2x = 22 Divide both sides by 2. x = 11 11 white buttons should be added to the bag.

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**Theoretical Probability**

Course 3 10-4 Theoretical Probability Check It Out: Example 3 Stephany has 2 dimes and 3 nickels. How many pennies should be added so that the probability of drawing a nickel is ? 3 7 Adding pennies to the bag will increase the number of possible outcomes. Let x equal the number of pennies. 3 5 + x = 7 Set up a proportion. 3(5 + x) = 3(7) Find the cross products.

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**Check It Out: Example 3 Continued**

Course 3 10-4 Theoretical Probability Check It Out: Example 3 Continued 15 + 3x = 21 Multiply. – – 15 Subtract 15 from both sides. 3x = 6 Divide both sides by 3. x = 2 2 pennies should be added to the bag.

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**Theoretical Probability**

Course 3 10-4 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 3 10-4 Theoretical Probability Example 4: Find the Probability of Mutually Exclusive Events 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? The event “total = 2” consists of 1 outcome, (1, 1), so P(total = 2) = . 1 36 1 36 = P(game ends) = P(total = 2) The probability that you will lose is , or about 3%. 1 36

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**Theoretical Probability**

Course 3 10-4 Theoretical Probability Check It Out: Example 4 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 3 10-4 Theoretical Probability Check It Out: Example 4 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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**Theoretical Probability Insert Lesson Title Here**

Course 3 10-4 Theoretical Probability Insert Lesson Title Here 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 3’s) 4. P(total shown > 10) 1 2 1 2 1 36 1 12

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