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Theoretical Probability

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  1. Theoretical Probability Goal: to find the probability of an event using theoretical probability.

  2. Probability • A probability experiment is an activity in which results are observed. Each observation is called a trial, and each result is called an outcome. The sample space is the set of all possible outcomes of an experiment.

  3. An Event… • An event is any set of one or more outcomes. • The probability of an event, written P(event), is a number from 0 (or 0%) to 1 (or 100%) that tells you how likely the event is to happen. • A probability of 0 means the event is impossible, or can never happen. • A probability of 1 means the event is certain, or has to happen. • The probabilities of all possible outcomes in the sample space add up to 1.

  4. Theoretical probability • Theoretical probability is used to estimate probabilities by making certain assumptions about an experiment. • The assumption is that all outcomes that are equally likely, that is, they all have the same probability. • To find theoretical probability: number of outcomes in the event total possible outcomes

  5. Example #1: • An experiment consists of rolling a fair die. There are 6 possible outcomes: 1, 2, 3, 4, 5, and 6. • What is the probability of rolling a 3, P(3)? • What is the probability of rolling an odd number, P(odd number)? • What is the probability of rolling a number less than 5, P(less than 5)?

  6. Example #2: • An experiment consists of rolling one fair die and flipping a coin. • Show a sample space that has all outcomes equally likely.

  7. What is the probability of getting tails, P(tails)? • What is the probability of getting an even number and heads, P(even # and heads)? • What is the probability of getting a prime number, P(prime)?

  8. Example #3: • Using the spinner, find the probability of each event. • P(spinning A) • P(spinning C) • P(spinning D)

  9. Example #4: • Suppose you roll two fair dice and are considering the sum shown. • Since each die represents a unique number, there are 36 total outcomes. • What is the probability of rolling doubles? • There are 6 outcomes of rolling doubles: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), and (6, 6). • P(double) =

  10. Example #4 cont…. • What is the probability of rolling a total of 10? • There are 6 outcomes in the event “a total of 10”: (4, 6), (5, 5), and (6, 4). • P(total = 10) = • What is the probability that the sum shown is less than 5? • There are 6 outcomes in the event “total < 5” :(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), and (3, 1). • P(total < 5) =

  11. Example #5: An experiment consists of rolling a single fair die. • Find P(rolling an even number) • Find P(rolling a 3 or a 5)

  12. Example #6: Mrs. Rife has candy hearts in a jar. She has 27 red hearts, 16 pink hearts, and 17 white hearts. • What is the theoretical probability of drawing a white heart? A red heart?.

  13. Example #7: Find each probability. • P(white) • P(red) • P(green) • P(black) • What should be the sum of the probabilities? Is it true?

  14. Example #8: Three fair coins are tossed: a penny, a dime, and a quarter. Copy the table and then find the following • P(HHT) • P(TTT) • P(0 tails) • P(1 tail) • P(2 heads) • P(all the same)

  15. Example #7: Find each probability. • P(white) • P(red) • P(green) • P(black) • What should be the sum of the probabilities? Is it true?