2Bernoulli TrialsThe basis for the probability models we will examine in this chapter is the __________________.We have Bernoulli trials if:there are _____ possible outcomes (___________ and ____________).the probability of success, p, is ____________.the trials are ______________.
3The Geometric ModelA single Bernoulli trial is usually not all that interesting.A Geometric probability model tells us the probability for a random variable that counts the number of Bernoulli trials until the __________________.Geometric models are completely specified by one parameter, _______________________, and are denoted Geom(__).
4The Geometric Model (cont.) Geometric probability model for Bernoulli trials: Geom(p)p = probability of successq = 1 – p = probability of failureX = # of trials until the first success occursP(X = x) =
5IndependenceOne of the important requirements for Bernoulli trials is that the trials be _________________.When we don’t have an infinite population, the trials are __________________. But, there is a rule that allows us to pretend we have _________________ trials:The ___% condition: Bernoulli trials must be independent. If that assumption is violated, it is still okay to proceed as long as the sample is smaller than ____% of the population.
6TI Tips Under Distributions you will find both geometpdf and geometcdf Use geometpdf(p,x) to find the probability of any ________ outcome. p is the individual probability of success and x is the number of trials until you get a success. Ex. geometpdf(.2,5) ≈Use geometcdf(p, x) to find the probability of success ________________ the xth trial.Ex. Geometcdf(.2, 4) ≈
7Ex. Approximately 3.6% of all (untreated) Jonathan apples had bitter pit in a study conducted by the botanists Ratkowsky and Martin (Australian Journal of Agricultural Research, Vol. 23, pp ). Bitter pit is a disease of apples resulting in a soggy core, which can be caused either by over watering the apple tree or by a calcium deficiency in the soil. Let n be a random variable that represents the first Jonathan apple chosen at random that has bitter pit.a) Find the probability that n = 3, n = 5, n= 12.b) Find the probability that n < 5.c) Find the probability that n > 5.d) State the mean and standard deviation.
10The Binomial ModelA Binomial model tells us the probability for a random variable that counts the number of ____________ in a fixed number of __________ _________.Two parameters define the Binomial model: ___, ___________________; and, ___, __________ _______________. We denote this Binom(n, p).
11The Binomial Model (cont.) In n trials, there areways to have k successes.Read nCk as “______________.”Note: n! = n x (n-1) x … x 2 x 1, and n! is read as “________________.”(Can also be shown as)Find the following.a) b)10C4
12The Binomial Model (cont.) Binomial probability model for Bernoulli trials: Binom(n,p)n =p =q = =X =P(X = x) =
13TI tips Under Distributions you will find both binompdf and binomcdf Use binompdf(n,p,x) to find the probability of any ________ outcome. n is the number of trials, p is the individual probability of success and x is the number of successes. Ex. binompdf(5,.2,2) ≈Use binomcdf(n,p, x) to find the probability of getting _____________________successes in n trials.Ex. binomcdf(5,.2, 2) ≈
14Ex. A basketball player makes 70% of his freethrows Ex. A basketball player makes 70% of his freethrows. Assuming the shots are independent, what is the probability of each of the following?a) He makes exactly 3 out of the next 5 attempts.b) He makes at most 3 out of the next 5.c) He makes at least 3 out of the next 10.
15The Normal Model to the Rescue As long as the ________________ Condition holds, we can use the Normal model to approximate Binomial probabilities.Success/failure condition: A Binomial model is approximately Normal if we expect at least ___ successes and ____ failures:
16Continuous Random Variables When we use the Normal model to approximate the Binomial model, we are using a ___________ random variable to approximate a __________ random variable.So, when we use the Normal model, we no longer calculate the probability that the random variable equals a ___________ value, but only that it lies _____________ two values.
17Ex. An orchard owner knows that he’ll have to use about 6% of the apples he harvests for cider because they will have bruises or blemishes. He expects a tree to produce about 300 apples.a) Verify that you can use the normal model to approximate the distribution.b) Find the probability there will be no more than a dozen cider apples.c) Is it likely there will be more than 50 cider apples? Explain.
18What Can Go Wrong? Be sure you have Bernoulli trials. You need ____ outcomes per trial, a constant _________________, and _______________.Remember that the ______________ provides a reasonable substitute for _______________.Don’t confuse ___________and _____________ models.Don’t use the Normal approximation with ___________.You need at least ___ successes and ____ failures to use the Normal approximation.
19What have we learned? Geometric model Binomial model Normal model When we’re interested in the number of Bernoulli trials __________________________.Binomial modelWhen we’re interested in the number of _________ in a certain number of Bernoulli trials.Normal modelTo ______________ a Binomial model when we expect at least ___ successes and ___ failures.