# Completed 4.2 Binomial Distribution F23

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1 Section 4.2 Binomial Distributions A. Binomial Experiments Definition: A binomial experiment is a probability experiment that satisfies these conditions. 1. The experiment has a ______________________ number of trials, where each trial is independent of other trials. 2. There are only _____________ possible outcomes of interest for each trial. Each outcome can be classified as _________________________ or __________________________. 3. The probability for success is ______________________________ for each trial. 4. The random variable ࠵? counts the number of successful trials. Notation for Binomial Experiments ___________: the number of trials ___________: the probability of success in a single trial ___________: the probability of failure in a single trial ___________: the random variable that represents a count of the number of successes is ࠵? trials Note: Success __________________________ always mean that something good occurred! Ex. 1: From a standard deck of cards, you pick a card, note whether or not the card is a club, and then replace the card. This experiment is repeated 5 times. Does this represent a binomial experiment? If so, find ࠵?, ࠵?, ࠵?, and list all possible values of ࠵?. If it is not a binomial experiment, explain why. fixed 2 a success a failure the same n p 9 i p g does not Binomial Clubs or Not Clubs n S p 132 4 25 9 1 p 1 25 075 x of trials successful x 0 I 2,3 4,5
2 B. Binomial Probability Formula In a binomial experiment, the probability of exactly ࠵? successes in ࠵? trials is given by the formula ࠵?(࠵?) = ࠵?(࠵?, ࠵?)࠵? ࠵? ௡ି௫ = ࠵?! (࠵? − ࠵?)! ࠵?! ࠵? ࠵? ௡ି௫ . The number of failures is ࠵? − ࠵?. Ex. 2: Rotator cuff surgery has a 90% chance of success. The surgery is performed on three patients. Find the probability of the surgery being successful on exactly two patients. C. Mean, Variance, and Standard Deviations of Binomial Probability Distributions By listing the possible values of ࠵? with the corresponding probabilities, we can construct a binomial probability distribution. If ࠵? is a binomial random variable, ࠵? is the total number of trials, ࠵? is the probability of success and ࠵? is the probability of failure, then Expected Value: ࠵?(࠵?) = ࠵? = ࠵?࠵? Variance: ࠵?(࠵?) = ࠵? = ࠵?࠵?࠵? Standard Deviation: ࠵?࠵?(࠵?) = ࠵? = ࠵?࠵?࠵? nCxpxqn X p 90 g 10 n 3 x 2 P x R Cx p g n x PIX 302 x 1.907 x C 10 0.2430 F S F S Not
3 Ex. 3: Thirty-nine percent of U.S. adults have very little or no confidence in newspapers. You randomly select eight U.S. adults. Find the probability that the number who have very little or no confidence in the newspapers is a. Exactly six. b. Exactly three. Ex. 4: The probability of success in an experiment is 0.65. If the experiment is conducted 10 times, what is the probability of at least six successes? Ex. 5: Fourteen percent of consumers have tried to purchase clothing second-hand rather than new in the past year. You randomly select 11 consumers. Find the probability that the number who have tried to purchase second-hand rather than new clothing is a. At most five b. More than three p 39 9 1 39 61 x 6 n 8 PIG Cake 39 6172 0367 or 037 x 3 P 3 g Cz x 1.3973 x C 6175 0.281 p 0.65 9 1 p 35 n 10 X Z 6 X 26 P 6 P 7 PC8 PIG P IO 10667 1.65767 61.3574 10 C C 65771.3573 to Ca 1 65781.355 local 65791.35 t 10C to C 657101.35 0 2377 0.2522 0.1757 0.0725 0.0135 I 0.7516 0 n 11 p 14 9 II is s PIX I 5 PCO PCI P 12 P 3 P 14 PCS p Col 14701.86 t 1 C C 14711.86310 t C2 C 14721.8679 n Cz 14731.867 1 Cy l 14741.8677 n Cs C 74751.8636 1903 t 3408 t 2774 1 355 t 0441 t 0101 I 99820 x 7 3 PIX 3 P 4 P 15 P 6 P1 P 8 TP 9 P IO P Il FX 3 1 P x 3 1 Plo PCI P 2 P 133 1 in Col 14701.86 ti C C 14 86110 1 C2 C 14721.8679 1636.143686