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Stats

Stats. Section 3.1-3.2 Notes. Rare Event Rule for Inferential Statistics.

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Stats

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  1. Stats Section 3.1-3.2 Notes

  2. Rare Event Rule for Inferential Statistics • If, under a given assumption (such as a lottery being fair), the probability of a particular observed event (such as five consecutive lottery wins) is extremely small, we conclude that the assumption is probably not correct.

  3. Fundamentals of Probability • An event is any collection of results or outcomes of a procedure. • A simple event is an outcome or an event that cannot be further broken down into smaller components. • The sample space for a procedure consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further.

  4. Notation for Probability • P denotes probability. • A, B, and C denote specific events. • P(A) denotes the probability of event A occurring.

  5. Rule 1: Relative Frequency Approximation of Probability • Conduct, or observe, a procedure a large number of times, and count the number of times that event A actually occurs. Based on these actual results, P(A) is estimated as follows:

  6. Rule 2: Classical Approach to Probability (Requires equally likely outcomes) • Assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occurring. If event A can occur in s of these n ways, then

  7. Rule 3: Subjective Probabilities • P(A), the probability of event A, is found by simply guessing or estimating its value based on knowledge of the relevant circumstances.

  8. Examples of the three different approaches to probability • Rule 1:Trying to find the probability that a person, when called on the phone, will first say “hello.” The procedure of calling must be repeated to obtain the needed probability.

  9. Examples of the three different approaches to probability • Rule 2: Trying to find the probability of drawing a “club” from a well-shuffled deck of 52 playing cards. Each of the four suits is equally likely in such a situation.

  10. Examples of the three different approaches to probability • Rule 3: Trying to determine the probability of having a snow day tomorrow. No doubt, a complete estimate!

  11. Law of Large Numbers • As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability.

  12. Examples • 1. Determine the probability that an adult that is at least 35 years old has three or more children. • In order to determine this probability it would be necessary to survey numerous adults that are at least 35 years of age. Let’s assume that of 1,054 surveyed adults of at least 35 years of age, it was determined that 274 had at least three children. In this case the relative frequency approach would be put to use to determine that

  13. Examples • Find the probability that when a couple has three children, they will have at least two girls. Assume that boys and girls are equally likely. • The sample space is needed here. The sample space is {b-b-b, b-b-g, b-g-b, b-g-g, g-b-b, g-b-g, g-g-b, g-g-g}. Since these outcomes are equally likely, classical approach may be used. So,

  14. Things that go without saying but worth noting • The probability of an impossible event is 0. • The probability of an event that is certain to occur is 1. • For any event A,

  15. Complement • The complement of event A, denoted by

  16. Rounding Off Probabilities • When expressing the value of a probability, either give the exact fraction or decimal or round off final results to three significant digits. All digits in a number are significant except for zeros that are included for proper placement of the decimal point.

  17. Example • 1. The probability 0.0034665 has five significant digits, and should be rounded to 0.00347. • 2. The probability of obtaining a “club” from a deck of cards can be expressed as or 0.25. Because this decimal is exact, there is no need to express it as 0.250.

  18. Odds • The odds in favor of event A occurring are the ratio • The actualodds against event A occurring are the ratio • The payoff odds against event A represent the ratio of net profit (if you win) to the amount bet.

  19. Example • Assume that you bet $20 on the number 21 in roulette, your probability of winning is

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