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M ARIO F . T RIOLA

S TATISTICS. E LEMENTARY. Chapter 3 Probability . M ARIO F . T RIOLA. E IGHTH. E DITION. Rare Event Rule for Inferential Statistics:

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M ARIO F . T RIOLA

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  1. STATISTICS ELEMENTARY Chapter 3 Probability MARIO F. TRIOLA EIGHTH EDITION

  2. 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.

  3. 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 Definitions

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

  5. 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

  6. 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

  7. 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

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

  9. 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

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

  11. The sample space consists of two simple events: the person is struck by lightning or is not. Using the relative frequency approximation (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.

  12. 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?

  13. 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

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

  15. 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? 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(woman) = = 0.6

  16. Random Variable a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure Probability Distribution a graph, table, or formula that gives the probability for each value of the random variable Definitions

  17. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Table 4-1 Probability DistributionNumber of Girls Among Fourteen Newborn Babies x P(x) 0.000 0.001 0.006 0.022 0.061 0.122 0.183 0.209 0.183 0.122 0.061 0.022 0.006 0.001 0.000

  18. Discrete random variable has a finite (or countable) number of values. Continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale with no gaps or interruptions. Definitions

  19. Probability Histogram Figure 4-3

  20. P(x) = 1 where x assumes all possible values 0 P(x) 1 for every value of x Requirements for Probability Distribution 

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