Chapter 11 Bernoulli Trials 11 1 The Binomial Distribution In the previous chapter we learned about i i d trials Recall that there are three ways we can have i i d trials 1 Our units are trials and we have decided to assume that they are i i d 2 We have a finite population and we will select our sample of its members at random with replacement the dumb form of random sampling The result is that we have i i d random variables which means the same thing as having i i d trials 3 We have a finite population and we have selected our sample of its members at random without replacement the smart form of random sampling If n N the ratio of sample size to population size is 0 05 or smaller then we get a good approximation if we treat our random variables as i i d In this chapter we study a very important special case of i i d trials called Bernoulli trials If each trial has exactly two possible outcomes then we have Bernoulli trials For convenient reference I will now explicitly state the assumptions of Bernoulli trials Definition 11 1 The assumptions of Bernoulli trials If we have a collection of trials that satisfy the three conditions below then we say that we have Bernoulli trials 1 Each trial results in one of two possible outcomes denoted success S or failure F 2 The probability of a success remains constant from trial to trial and is denoted by p Write q 1 p for the constant probability of a failure 3 The trials are independent We will use the method described on page 168 of Chapter 8 to assign the labels success and failure When we are involved in mathematical arguments it will be convenient to represent a success by the number 1 and a failure by the number 0 Finally 255 We are not interested in either of the trivial cases in which p 0 or p 1 Thus we restrict attention to situations in which 0 p 1 One reason that Bernoulli trials are so important is that if we have Bernoulli trials we can calculate probabilities of a great many events Our first tool for calculation is the multiplication rule that we learned in Chapter 10 For example suppose that we have n 5 Bernoulli trials with p 0 70 The probability that the Bernoulli trials yield four successes followed by a failure is P SSSSF ppppq 0 70 4 0 30 0 0720 Our next tool is extremely powerful and very useful in science It is the binomial probability distribution Suppose that we plan to perform observe n Bernoulli trials Let X denote the total number of successes in the n trials The probability distribution of X is given by the following equation n P X x px q n x for x 0 1 n 11 1 x n x Equation 11 1 is called the binomial probability distribution with parameters n and p it is denoted by the Bin n p distribution I will illustrate the use of this equation below a compilation of the Bin 5 0 60 distribution I replace n with 5 p with 0 60 and q with 1 p 0 40 Below I will evaluate Equation 11 1 six times for x 0 1 5 You should check a couple of the following computations to make sure you are comfortable using Equation 11 1 but you don t need to verify all of them P X 0 5 0 60 0 0 40 5 1 1 0 01024 0 01024 0 5 5 0 60 1 0 40 4 5 0 60 0 0256 0 07680 1 4 5 0 60 2 0 40 3 10 0 36 0 064 0 23040 P X 2 2 3 5 P X 3 0 60 3 0 40 2 10 0 216 0 16 0 34560 3 2 5 0 60 4 0 40 1 5 0 1296 0 40 0 25920 P X 4 4 1 5 P X 5 0 60 5 0 40 0 1 0 07776 1 0 07776 5 0 Whenever probabilities for a random variable X are given by Equation 11 1 we say that X has a binomial probability sampling distribution with parameters n and p and write this as X Bin n p There are a number of difficulties that arise when one attempts to use the binomial probability distribution The most obvious is that each trial needs to give a dichotomous response Sometimes it is obvious that we have a dichotomy For example if my trials are shooting free throws or attempting golf putts then the natural response is that a trial results in a make or miss Other P X 1 256 times the natural response might not be a dichotomy but the response of interest is For example in my example of the American roulette wheel in Chapter 10 the natural response is the winning number but if I like to bet on red then the response of interest has possible values red and not red Similarly in the game of craps I might be primarily interested in whether or not my come out results in a pass line win a dichotomy Thus let s suppose that we have a dichotomous response The next difficulty is that in order to calculate probabilities we need to know the numerical values of n and p Almost always n is known to the researcher and if it is unknown we might be able to salvage something by using the Poisson distribution which you will learn about in Chapter 13 There are situations in which p is known including the following Mendelian inheritance my roulette example above assuming the wheel is fair and my craps example assuming both dice are fair and they behave independently In other words sometimes I feel that the phenomenon under study is sufficiently well understood that I feel comfortable in my belief that I know the numerical value of p Obviously in many situations I won t know the numerical value of p For shooting free throws or attempting golf putts I usually won t know exactly how skilled the player golfer is When my interest is in a finite population typically I won t know the composition of the population hence I won t know the value of p Before I explore how to deal with the difficulty of p being unknown let s perform a few computations when p is known Example 11 1 Mendelian inheritance with the 3 1 ratio If you are unfamiliar with Mendelian inheritance you can find an explanation of the 1 2 1 3 1 and 9 3 3 1 ratios at http en wikipedia org wiki Mendelian inheritance Each trial offspring will possess either the dominant success or recessive failure phenotype Mendelian inheritance tells us that these will occur in the ratio 3 1 which means that p 3q 3 1 p Solving for p this equation becomes 4p 3 and finally p 0 75 Suppose that we will observe …
View Full Document
Unlocking...