Section 6.3 Binomial Distributions

A Gaggle of Girls Let’s use simulation to find the probability that a couple who has three children has all girls. P(girl) = 0.5 Let 0 = boy and 1 = girl. Use your calculator to choose 3 random digits to simulate this experiment. Complete this experiment 50 times in your group and record. Create a probability distribution for X = number of girls.

Gaggle continued What was your group’s probability for having three girls? Use your knowledge of probabilities to find the actual chance that a family with three children has three girls. Are these close?

Children, Again??? Two types of scenarios: A couple is going to have children until they have a girl. A couple is going to have children until they have a girl. Here, the random variable is how many children will it take to get a girl. A couple is going to have 3 children and we’ll count how many are girls. A couple is going to have 3 children and we’ll count how many are girls. Here, the random variable is how many girls there are out of the 3 children.

Dichotomous Outcomes Both of those situations have dichotomous (two) outcomes. Other examples with two outcomes: Coin toss (heads or tails) Coin toss (heads or tails) Shooting free throws (make or miss) Shooting free throws (make or miss) A game of baseball (win or lose) A game of baseball (win or lose)

Special Type of Setting In this chapter, we’ll study a setting with two outcomes where there are a fixed number of observations (or trials). The binomial distribution is a special type of setting in which there are two outcomes of interest.

4 Conditions for a Binomial Setting 1.There are two outcomes for each observation, which we call “success” or “failure.” 2.There is a fixed number n of observations. 3.The n observations are all independent. 4.The probability of success, called p, is the same for each observation.

Binomial Random Variables Binomial random variable: In a binomial setting, the random variable X = # of success. The probability distribution of X is called a binomial distribution. The parameters of a binomial distribution are n (the number of observations) and p (the probability of success on any one observation). The parameters of a binomial distribution are n (the number of observations) and p (the probability of success on any one observation). B(n, p) Is a binomial random variable discrete or continuous? Discrete…

Example Blood type is inherited. If both parents have the genes for the O and A blood types, then each child has probability 0.25 of getting two O genes and thus having type O blood. Is the number of O blood types among this couple’s 5 children a binomial distribution? If so, what are n and p ? If so, what are n and p ? If not, why not? If not, why not?

Example Deal 10 cards from a well-shuffled deck of cards. Let X = the number of red cards. Is this a binomial distribution? If so, what are n and p ? If so, what are n and p ? If not, why not? If not, why not?

Using the Calculator to Find Binomial Probabilities Under 2 nd VARS (DISTR), find 0:binompdf( This command finds probabilities for the binomial probability distribution function. The parameters for this command are binomialpdf(n, p, x) IN THAT ORDER. This will only give you the probability of a single x value. This will only give you the probability of a single x value.

Example Let’s go back to the couple having three children. Let X = the number of girls. p = P(success) = P(girl) = 0.5 The possible values for X is 0, 1, 2, 3. Using the binompdf(n,p,x) command, complete the probability distribution. What is the probability that the couple will have no more than 1 girl?

Cumulative Distribution Function The pdf command lets you find probabilities for ONE value of X at a time. binomialcdf(n, p, x) This time, you will be given the sum of the probabilities ≤ x. Be sure you remember this when answering a question This time, you will be given the sum of the probabilities ≤ x. Be sure you remember this when answering a question The cdf command finds cumulative probabilities. We can use it to quickly find probabilities such as P(X < 7) or P(X ≥ 4).

Corinne’s Free Throws Corinne makes 75% of her free throws over the course of a season. In a key game, she shoots 12 free throws and makes 7 of them. Is it unusual for her to shoot this poorly or worse? What is the probability that Corinne makes at least 6 of the 12 free throws?

Homework Chapter 6# 69-72, 86, 94