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Ch 14: From Randomness to Probability

Learn the basics of probability, including random phenomena, trials, outcomes, and events. Explore the Law of Large Numbers and formal probability rules. Practice with real-life examples of polling organizations contacting households.

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Ch 14: From Randomness to Probability

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  1. Ch 14: From Randomness to Probability Beginning of Unit 4, AP Stats

  2. Vocab! • Random phenomenon (“rolling a die”) • Has set of trials (“roll 20 times”) • Each trial has an outcome (“this time I rolled a 1”) • Outcomes combine to make an event (“Rolling a 4” --all times rolling a 4 fits into this category) • Try another example: List each item for a coin toss. Random phenomenon: Set of Trials: Outcome: Event:

  3. Law of Large Numbers • As n grows, collected data will reflect a probability • Probability = long-run (or large n) relative frequency of repeated independent events Caution! There is NOT a law of averages based on small n • Just because you know P, doesn’t mean you will see it within a small n

  4. Personal Probability • Subjective probability • Not based on long-run relative frequencies • Personal opinion • NOT probability! Be careful not to use “probability” this way!

  5. Formal Probability Rules • A = event; P(A) is between 0 and 1 (inclusive) • “Something Has to Happen Rule”: S = set of all possible outcomes, P(S) = 1; something in the set has to happen • Complement Rule: AC = complement of A P(A) = 1 – P(AC) If all Ps don’t add to 1, then Ps are not legitimate. 4. Addition Rule: P(A or B) = P(A) + P(B), assuming A and B are disjoint / mutually exclusive 5. Multiplication Rule:P (C and D) = P(C) x P(D), assuming C and D are independent (Independence Assumption) Note: disjoint events cannot be independent Example: A = get an A in class. B = get a B in class. A and B are disjoint. If I get an A, I know I am not getting a B, so they are not independent!

  6. Example Polling org.s contact their respondents by telephone. Random phone numbers are generated, and interviewers call those households. This method reaches about 76% of US households. Each household is independent of the others. • What is the probability that the interviewer will successfully contact the next household on the list? • What is the probability that the interviewer successfully contacts both of the next two households on the list? • What is the probability that the interviewer’s first successful contact will be the third household on the list? • What is the probability that the interviewer will make at least one successful contact among the next 5 households on the list?

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