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Theoretical Probability

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jersey
15 diapositives · 390 Views · Jul 28, 2014

This guide explores the concept of theoretical probability, focusing on how to calculate the probability of events based on assumed outcomes. It explains key terminology, including probability experiments, trials, outcomes, and events. Real-life examples—such as rolling dice, flipping coins, and drawing from a jar of candies—illustrate how to compute probabilities for varied scenarios. Theoretical probability estimates are made under the assumption that all outcomes are equally likely, providing a foundational understanding for calculating and interpreting event probabilities.

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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?