1. Section 6.3 Day 1 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. 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.

5. Binary Outcomes Both of those situations have binary (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)

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 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.

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). 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 so, what are n and p ? If not, why not? 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 so, what are n and p ? If not, why not? If not, why not?

11. 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.

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 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. Each child born to a particular set of parents has probability of 0.25 having blood type O. Suppose these parents have 5 children. Let X = number of children who have type O blood. Then X is B(5, 0.25). What is the probability that exactly 2 of the children have type O blood? What is the probability that exactly 2 of the children have type O blood? Make a table for the pdf of the random variable X. Then use the calculator to find the probabilities of all possible values X, and complete the table. Make a table for the pdf of the random variable X. Then use the calculator to find the probabilities of all possible values X, and complete the table. Verify that the sum of the probabilities is 1. Verify that the sum of the probabilities is 1. Construct a histogram of the pdf. Construct a histogram of the pdf.

16. Binomial Formula Now, we’ll learn about the basis for the binomial calculations: the binomial formula. Notice the = mark. This is a combinatorial. It is read “n choose k.” Use n C r in your calculator! p stands for the probability of success. n represents the number of observations. k is the value of x of which you’re asked to find the probability.

17. Completing the Formula At the beginning of class, we were trying to find out the probability that a couple who has three children has one girl. The formula, then, is

18. Try This… The number of switches that fail inspection follows a binomial distribution with n = 10 and p = 0.1. Find the probability that no more than 1 switch fails.

19. Why can’t I just use binomialpdf or binomialcdf ??? That is fine for multiple choice or to check your answer. On free response, judges expect you to at least be able to fill in the formula, but then you can use pdf or cdf to get the value. Find it on your formula sheet.

20. Example Patti is given a 10-item multiple choice quiz, each question with 5 answer choices. She randomly guesses on each of the questions. X = the number of correct guesses. Show that X is a binomial distribution. Show that X is a binomial distribution. Find P(X = 3) using the formula. Explain what it means. Find P(X = 3) using the formula. Explain what it means. Patti must get at least six questions correct to get a passing score. Fill in the formula for Patti earning a passing score and then calculate the probability. Patti must get at least six questions correct to get a passing score. Fill in the formula for Patti earning a passing score and then calculate the probability.

21. The Mean and Standard Deviation of a Binomial Random Variable These are the formulas for a binomial distribution ONLY. Be sure you are looking at a binomial random distribution before you make the calculations. These formulas are on your formula sheet. Let’s locate them.

22. Example If there are 10 multiple choice questions on a test, and each question has 4 answer choices, how many on average will a student get right by purely guessing? What is the standard deviation? What is the standard deviation? Note: This is much easier to use since we could have hundreds of observations to look at. It would not be advised to spend the time making your own probability distribution. Note: This is much easier to use since we could have hundreds of observations to look at. It would not be advised to spend the time making your own probability distribution.

23. Homework p. 403 (70, 72, 75, 76, 77, 78, 81)