AP Test Review #3 Focus  Binomial and Geometric Distributions  Basic Probability  Tree Diagrams

Bernoulli Trial A distribution where:  The outcomes are success or failure  Each trial is independent  The probability of success is constant

Binomial and Geometric Distributions Binomial and geometric distributions are Bernoulli trials Binomial Dist. – the probability of a certain number of successes out of a set number of trials (n,p,k) Geometric Dist. – the probability that the 1 st success occurs on the kth try. (p,k)

Binomial Formulas Probability of k success out of n trials (pdf) Probability of k or fewer successes out of n trials – use binomial cdf! Mean: Standard Deviation:

Geometric Formulas Probability of 1 st success on the kth trial Cdf = probability of 1 st success occurs on or before the kth trial Mean:

Example 1: Assume the probability of getting a blue token on each turn in a certain game is constant and equal to Find: a)The probability of getting 4 blue tokens in 7 turns. b)The probability of getting at least 4 blue tokens in 7 turns. c)Mean and standard deviation for the number of blue tokens in 11 turns. d)The probability the 1 st blue token occurs on the 5th turn. e)The probability that it takes less than 4 turns to get the 1 st blue token f)The mean number of turns to get the 1 st blue token.

Answers a)P = b)P = c) = 3.85 and  = d)P = e)P =.7254 f) = 2.857

Basic Probability formulas Union (or) Conditional (limit the total) Complement

Example 2 The probability of going to the junior Prom is 47% and the probability of going to the senior Ball is 39%. In addition, the probability of going to both the Junior Prom and Senior Ball is 18%. Find a)P(junior prom or senior ball) b)P(junior prom|senior ball) c)P(not going junior prom nor senior ball) d)P(going to senior ball but not junior prom) e)P(senior ball|junior ball)

Answers a) =.69 b).18/.39 =.462 c) d) =.21 e).18/.47 =.383

Last Example The probability that a Freshman entering college goes straight to a 4-year school is 38%. The probability that a student that goes straight to a 4-year school graduating in 4 years is 27%. A Freshman entering college that goes to a 2-year school to start and graduating in 4 years is 19%. a) Find the probability of graduating in 4 years. b) Find the probability of a student starting at a 2-year college and graduating in 4 years.