7.5 Binomial and Geometric Distributions *two of more commonly found discrete probability distributions Thursday, June 30, 2016

Properties of Binomial Experiment: 1)2 possible values for each observation or trial, success (s) and fail (f) 2)Experiment has a fixed number of trials (n) 3)Outcomes of different trials are independent 4)Probability that a trial results in a success is the same for each trial. Binomial random variable is defined as: x = number of successes Probability distribution of x is called the binomial probability distribution

Recall hot tub model selection 4 customers – fixed n Electric vs. gas – 2 possible outcomes s – electric f – gas Each customer choice independent of others Probability of choosing electric (s) is same for each customer

Recall that there were four ways to get one s (E) SFFF, FSFF, FFSF, FFFS and prob.: (.6) 3 (.4) of each no matter the order of the S’s and F’s x – number of successes So p(x=1) = 4(.6) 3 (.4) P(x) = (number of ways)(S) x (F) x-1

Where is probability of success and x is number of successes and n is the number of trials Combinations – count how many ways for you on calc Math to PRB #3 nCr Ex Computer Monitors pg. 383

Using binomial formula can be tedious unless n is small, we’ll use the calculator when n is larger, but show numbers plugged into form. ____+____+_____+...+_____ Binompdf (n,,x) (exactly that number) Binomcdf (n,, x) (an interval of numbers always goes to the left)

Sampling without replacement Trials aren’t independent so not binomial, but when n is much smaller than N, the binomial distribution can approximate the probabilities if At most 5% of the population is sampled Use binomial for approximate probability and calc to get it

Binomial dist. only symmetric when otherwise skewed right when Mean and standard deviation for binomial distribution use same formulas as general discrete r.v.’s but binomial has simpler versions using algebra and simplification Ex credit cards paid in full

Geometric Interested in number of trials that must be carried out before a success occurs. (other 3 properties are the same) Probability of failures and success Ex Jumper cables pg. 389

Example Assume that 13% of people are left-handed. If we select five people at random, find the probability of each outcome: a.The first lefty is the fifth person chosen b.The first lefty is the second or third person c.There are exactly three lefties in the group d.There are at least three lefties in the group e.There are no more than 3 lefties in the group

Binomial Example Suppose 20 donors come to a blood drive. Recall that 6% of people are “universal donors.” 1.What are the mean and standard deviation of the number of universal donors among them? 2.What is the probability that there are 2 or 3 universal donors?