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Types of Distributions Probablity Distribution (non specific) o o Binomial Distribution Probablities must add to 1 Expected Outcome (Mean of the distribution)—This has been on every AP exam I have seen, either on FRQ or MC o bg x xi pi B n, p o Multiply each outcome by its probability and add together Standard deviation (never been on AP exam that I know of) o b g x xi x pi 2 0 0.4 1 0.3 2 0.2 3 0.1 x (0)(0.4) (1)(0.3) 2(0.2) 3(01 . ) 10 . On average, Landry’s sells 1 hat per week. He should buy fifty two hats per year. Normal Distribution If there is reason to believe that a distribution is normal, you must state that it is normal and state the average ( ) and the standard deviation ( ) o This can be done by simply writing the shorthand version: You must draw a normal curve picture with the problem’s numbers referenced in it, and it would also be good to reference the formula for a standardized score (z-score) o b g N , x x or z z / n Calculate your probability, and verify with a calculator State your probability and its meaning in the context of the question Example: A box of candy is known to have an average of 50 pieces. If it is known that the amount of packaged candy is normally distributed with a standard deviation of 5, is it likely to get a box with 62 pieces? b g z 62 5 50 Pb X 62g Pb z 2.4g 0.0082 N 50,5 There is a 0.82% chance that a box would get 62 pieces of candy or more. So this is very unlikely. . Make sure to state what p represents. You must show that it meets the four criteria o Success/Failure o There are a set number (n) of observations o Probability of success never changes o Each observation is independent Plug into the formula o Example: The number of hats sold at Landry’s per week is as follows X P(X) You must state that it is binomial, state what a success is, what the probability of success is and how many observations are being made. o This can be done with the shorthand: b g b g P X k n Ck pk 1 p n k Describe you answer in the context of the question Example: The probability of making any money in a state lottery is 0.4. There is a drawing once a week. What is the probably that you would win at least six times in a seven week period? B(7, 0.4) where n is the number of weeks being observed and p = the probability of making any money = 0.4 1. success=making money/failure=not making money 2. There are seven weeks of observations 3. p = probability of winning = 0.4 4. There is no reason to believe that each drawing is not independent b g Pb X 6g Pb X 7g Cb 0.4gb 0.6g C b 0.4gb 0.6g P X6 6 7 1 6 7 7 0 7 0.0172 0.0016 0.0188 Geometric Distribution 1. 2. 3. 4. Everything is the same as for a binomial, except for rule #2 Same Observations are made until a success occurs Same Same Plug into formula o o b g b g p Pb X kg b 1 pg P X k 1 p k 1 k Example: Using the information about the lottery above. What is the probability that you would have to play at least 10 weeks until you got a win? b g b gb g P X 10 P X 9 0.6 0.01 9