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C.M. Pascual

S TATISTICS. Chapter 4a Probability. C.M. Pascual. 5-1 Overview 5-2 Fundamentals 5-3 Addition Rule 5-4 Multiplication Rule: Basics 5-5 Multiplication Rule: Complements and Conditional Probability 5-6 Probabilities Through Simulations 5-7 Counting. Chapter 5 Probability.

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C.M. Pascual

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  1. STATISTICS Chapter 4a Probability C.M. Pascual

  2. 5-1 Overview 5-2 Fundamentals 5-3 Addition Rule 5-4 Multiplication Rule: Basics 5-5 Multiplication Rule: Complements and Conditional Probability 5-6 Probabilities Through Simulations 5-7 Counting Chapter 5Probability

  3. Objectives develop sound understanding of probability values used in subsequent chapters develop basic skills necessary to solve simple probability problems Rare Event Rule for Inferential Statistics: If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. 5-1 Overview

  4. Event - any collection of results or outcomes from some procedure Simple event - any outcome or event that cannot be broken down into simpler components Sample space - all possible simple events 5-2 Fundamentals Definitions

  5. P - denotes a probability A, B, ...- denote specific events P (A)- denotes the probability of event A occurring Notation

  6. Rule 1: Relative Frequency Approximation Conduct (or observe) an experiment a large number of times, and count the number of times event A actually occurs, then an estimate of P(A) is Basic Rules for Computing Probability

  7. Rule 1: Relative Frequency Approximation Conduct (or observe) an experiment a large number of times, and count the number of times event A actually occurs, then an estimate of P(A) is Basic Rules for Computing Probability P(A) = number of times A occurred number of times trial was repeated

  8. Rule 2: Classical approach (requires equally likely outcomes) If a procedure has n different simple events, each with an equal chance of occurring, and s is the number of ways event A can occur, then Basic Rules for Computing Probability

  9. Rule 2: Classical approach (requires equally likely outcomes) If a procedure has n different simple events, each with an equal chance of occurring, and s is the number of ways event A can occur, then Basic Rules for Computing Probability s P(A) = number of ways A can occur = n number of different simple events

  10. Rule 3: Subjective Probabilities P(A), the probability of A, is found by simply guessing or estimating its value based on knowledge of the relevant circumstances. Basic Rules for Computing Probability

  11. Rule 1 The relative frequency approach is an approximation.

  12. Rule 2 The classical approach is the actual probability. Rule 1 The relative frequency approach is an approximation.

  13. As a procedure is repeated again and again, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability. Law of Large Numbers

  14. Illustration of Law of Large Numbers Figure 3-2

  15. The sample space consists of two simple events: the person is struck by lightning or is not. Because these simple events are not equally likely, we can use the relative frequency approximation (Rule 1) or subjectively estimate the probability (Rule 3). Using Rule 1, we can research past events to determine that in a recent year 377 people were struck by lightning in the US, which has a population of about 274,037,295. Therefore, P(struck by lightning in a year)  377 / 274,037,295 1/727,000 Example:Find the probability that a randomly selected person will be struck by lightning this year.

  16. There are 5 possible outcomes or answers, and there are 4 ways to answer incorrectly. Random guessing implies that the outcomes in the sample space are equally likely, so we apply the classical approach (Rule 2) to get: P(wrong answer) = 4 / 5 = 0.8 Example:On an ACT or SAT test, a typical multiple-choice question has 5 possible answers. If you make a random guess on one such question, what is the probability that your response is wrong?

  17. The probability of an impossible event is 0. The probability of an event that is certain to occur is 1. Probability Limits

  18. The probability of an impossible event is 0. The probability of an event that is certain to occur is 1. 0  P(A)  1 Probability Limits

  19. The probability of an impossible event is 0. The probability of an event that is certain to occur is 1. 0  P(A)  1 Probability Limits Impossible to occur

  20. The probability of an impossible event is 0. The probability of an event that is certain to occur is 1. 0  P(A)  1 Probability Limits Certain to occur Impossible to occur

  21. Possible Values for Probabilities Certain 1 Likely 0.5 50-50 Chance Unlikely Figure 3-3 Impossible 0

  22. Complementary Events

  23. The complement of event A, denoted by A, consists of all outcomes in which event A does not occur. Complementary Events

  24. The complement of event A, denoted by A, consists of all outcomes in which event A does not occur. P(A) (read “not A”) P(A) Complementary Events

  25. Example: Testing CorvettesThe General Motors Corporation wants to conduct a test of a new model of Corvette. A pool of 50 drivers has been recruited, 20 or whom are men. When the first person is selected from this pool, what is the probability of not getting a male driver?

  26. Because 20 of the 50 subjects are men, it follows that 30 of the 50 subjects are women so, P(not selecting a man) = P(man) = P(woman) = 30 = 0.6 50 Example: Testing CorvettesThe General Motors Corporation wants to conduct a test of a new model of Corvette. A pool of 50 drivers has been recruited, 20 or whom are men. When the first person is selected from this pool, what is the probability of not getting a male driver?

  27. give the exact fraction or decimal or Rounding Off Probabilities

  28. give the exact fraction or decimal or Rounding Off Probabilities • round off the final result to three significantdigits

  29. Odds

  30. actual odds in favor of event A are the reciprocal of the odds against that event, b:a (or ‘b to a’) Odds • actual odds against event A occurring are the ratio P(A) P(A), usually expressed in the form of a:b • (or ‘a to b’), where a and b are integers with no common factors

  31. Odds • The payoff odds against event A     represent the ratio of net profit (if you win)     to the amount of the bet. Payoff odds against event A = (net profit):(amount bet)

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