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Section 6.3

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.

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Section 6.3

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  1. Section 6.3 Binomial Distributions

  2. 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.

  3. 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?

  4. Children, Again??? • Two types of scenarios: • 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. • Here, the random variable is how many girls there are out of the 3 children.

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

  6. 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.

  7. 4 Conditions for a Binomial Setting • There are two outcomes for each observation, which we call “success” or “failure.” • There is a fixed number n of observations. • The n observations are all independent. • The probability of success, called p, is the same for each observation.

  8. 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). B(n, p) • Is a binomial random variable discrete or continuous? Discrete…

  9. 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 not, why not?

  10. 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 not, why not?

  11. Using the Calculator to Find Binomial Probabilities • Under 2nd 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.

  12. 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?

  13. 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 • The cdf command finds cumulative probabilities. We can use it to quickly find probabilities such as P(X < 7) or P(X ≥ 4).

  14. 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?

  15. Homework Chapter 6# 69-72, 86, 94

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