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Theoretical Probability

Theoretical Probability. 9.4. Pre-Algebra. 3. 1. 1. 2. 36. 6. 2. 36. 1. 5. 4. 9. 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. Learn to estimate probability using theoretical methods .

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Theoretical Probability

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  1. Theoretical Probability 9.4 Pre-Algebra

  2. 3 1 1 2 36 6 2 36 1 5 4 9 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. Learn to estimate probability using theoretical methods.

  4. Vocabulary theoretical probability equally likely fair mutually exclusive

  5. 1 5 x = 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

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

  7. 1 5 The spinner is fair, so all 5 outcomes are equally likely. The probability of spinning a 4 is P(4) = . Example: Calculating Theoretical Probability An experiment consists of spinning this spinner once. A. What is the probability of spinning a 4?

  8. 2 5 P(spinning an even number) = number of possible even numbers 5 = Example: 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.

  9. 3 5 P(spinning a number less than 4)= Example: 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.

  10. 1 5 The spinner is fair, so all 5 outcomes are equally likely. The probability of spinning a 1 is P(1) = . Try This An experiment consists of spinning this spinner once. A. What is the probability of spinning a 1?

  11. 3 5 P(spinning an odd number) = number of possible odd numbers 5 = Try This 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.

  12. 2 5 P(spinning a number less than 3)= Try This 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.

  13. Example: 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

  14. 6 1 12 2 P(tails) = = Example: 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).

  15. 3 1 12 4 P(even number and heads) = = Example: 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).

  16. 6 1 12 2 P(prime number) = = Example: 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).

  17. Try This 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

  18. 2 1 4 2 P(head and tail) = = Try This 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).

  19. 1 4 P(both heads) = Try This 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).

  20. 1 4 P(both tails) = Try This 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).

  21. 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 AorB will occur) = P(A) + P(B)

  22. Example: 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.

  23. 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 1 4 4 5 5 36 36 36 36 36 36 = + = The probability that the game will end is , or about 13.9%. Example Continued P(game ends) = P(total = 5) + P(total = 2)

  24. Try This 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.

  25. 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 2 4 2 4 4 4 = + = The probability that the game will end is , or 50%. Try This Continued P(game ends) = P(both tails) + P(both heads)

  26. 1 1 1 1 36 12 2 2 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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