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PROBABILITY RULES! Conditions for Valid Probability Models A probability must be a number between 0 and 1 (or 0% and 100%). 0 P(A) 1 The sum of all the probabilities in a sample space must equal 1 (100%). P( A) 1 sample space Probability Rules! Complement Event Rule: P(not A) = 1 – P(A) Addition Rule for “either/or”: P(A or B) = P(A) + P(B) – P(A and B) (general form) If A and B are disjoint events, then P(A or B) = P(A) + P(B) because P(A and B) = 0. Multiplication Rule for “and/both”: P(A and B) = P(A) . P(B/A) (general form) If A and B are independent events, then P(A and B) = P(A) . P(B) because P(B/A) = P(B). Conditional Probability Rule: P(B/A)= P(A and B) P(A) PROBABILITY RULES! Central Limit Theorem The sampling distribution of 𝑥̅ is approximately normal, centered at µ, 𝜎 the population mean, with standard deviation . 𝑛 √ Binomial Conditions Two Outcomes, “Success” and “Failure” Probability of Success = p Probability of Failure = 1 – p n = number of independent trials Binomial Random Variable Let X = number of successes in n independent trials, then P( x k ) nCk p k 1 p np nk np(1 p) If the expected number of successes and failures are both at least 10, then the binomial distribution is approximately normal.