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10-4

10-4. Theoretical Probability. Course 3. Warm Up. Problem of the Day. Lesson Presentation. Theoretical Probability. 10-4. 1. 1. 1. 2. 36. 2. 6. 12. 1. 5. 4. 9. Course 3. Warm Up 1. If you roll a number cube, what are the possible outcomes? 2. Add + .

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10-4

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  1. 10-4 Theoretical Probability Course 3 Warm Up Problem of the Day Lesson Presentation

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

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

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

  6. Theoretical Probability 10-4 1 5 x = Course 3 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

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

  8. Theoretical Probability 10-4 1 5 Course 3 Additional 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. number of outcomes for 4 5 P(4) = =

  9. Theoretical Probability 10-4 2 5 P(even number) = number of possible even numbers 5 = Course 3 Additional 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.

  10. Theoretical Probability 10-4 1 5 Course 3 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. number of outcomes for 1 5 P(1) = =

  11. Theoretical Probability 10-4 3 5 P(odd number) = number of possible odd numbers 5 = Course 3 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.

  12. Theoretical Probability 10-4 Course 3 Additional 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

  13. Theoretical Probability 10-4 6 1 12 2 P(tails) = = Course 3 Additional Example 2B: Calculating Theoretical Probability for a Fair Coin An experiment consists of flipping a coin. Find the probability of the event. P(tails) There are 6 outcomes in the event “flipping tails”: (1, T), (2, T), (3, T), (4, T), (5, T), and (6, T).

  14. Theoretical Probability 10-4 2 1 4 2 P(head and tail) = = Course 3 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).

  15. Theoretical Probability 10-4 1 4 P(both tails) = Course 3 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).

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

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

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

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

  20. Theoretical Probability 10-4 Course 3 • 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 AorB will occur) = P(A) + P(B)

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

  22. Theoretical Probability 10-4 Course 3 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.

  23. Theoretical Probability 10-4 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 1 1 1 1 1 4 4 4 2 2 4 = + = The probability that the game will end is , or 50%. Course 3 Check It Out: Example 4 Continued P(game ends) = P(both tails) + P(both heads)

  24. Theoretical Probability 10-4 1 1 1 1 36 12 2 2 Course 3 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)

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