1. Warm-up Grab a die and roll it 10 times and record how many times you roll a 5. Repeat this 7 times and record results. This time roll the die until you get a 5. Record how many rolls it took you. Repeat this 7 times and record results.

2. Warm-Up What’s the smallest number you can get for the first one? What’s the largest number? What’s the smallest number you can get for the second one? What’s the largest number?

3. 6.2 Homework Questions

4. Section 6.3 Binomial Random Variables

5. What does “bi” mean? List all of the words you can think of that start with “bi”…

6. Binomial Setting The four conditions for a binomial setting are: 1. Success/Failure 2. Independent Trials 3. Constant “p” (probability of success) 4. Set number of trials, n

7. Geometric The four conditions for a geometric setting are: 1. Success/Failure 2. Independent Trials 3. Constant “p” (probability of success) 4. No set number of trials, n

8. Binomial Random Variable The count X of successes in a binomial setting is a binomial random variable. The probability distribution of X is a binomial distribution with parameters n and p. The possible values of X are the whole numbers from 0 to n.

9. Binomial? Genetics says that children receive genes from each of their parents independently. Each child of a particular pair of parents has probability 0.25 of having type O blood. Suppose these parents have 5 children. Let X = the number of children with type O blood. Shuffle a deck of cards. Turn over the first 10 cards, one at a time. Let Y = the number of aces you observe. Shuffle a deck of cards. Turn over the top card. Put the card back in the deck, and shuffle again. Repeat this process until you get an ace. Let W = the number of cards required.

10. Binomial Probabilities Let’s do the children’s gene problem… P(none of the children have type O)= P(x=1)

11. Building the formula

12. Number of arrangements: Binomial Coefficient

13. For example… When x = 1, we had 5 arrangements… There is a button on your calculator! 5 nCr 1 Math – Prob – nCr

14. Binomial Probability

15. Example Find the probability that exactly 3 children have type O blood. Should the parents be surprised if more than 3 of their children have type O blood? Justify your answer.

16. Mean and Standard Deviation of a Binomial Distribution Blood Type Probability Distribution: X012345 P(X)0.237300.395510.263670.087890.014650.00098

17. Mean and Standard Deviation of Binomial Random Variables

18. Homework #3 Together, let’s do numbers 69-72 Pg. 403 (73-75, 77, 79, 80, 82, 84-87, 89-92, 94-105)

19. Warm-Up

20. Normal Approximation for Binomial Distributions

21. Example Suppose that exactly 60% of all adult US residents would say “agree” if asked if they think shopping is frustrating. A survey asked nationwide sampled 2500 adults. Let X = the number of people who agree. ◦ Show that X is approximately a binomial random variable.

22. Example Check the conditions for using a Normal approximation in this setting.

23. Example Use a Normal distribution to estimate the probability that 1520 or more of the sample agree.

24. Homework #3 Together, let’s do numbers 69-72 Pg. 403 (73-75, 77, 79, 80, 82, 84-87, 89-92, 94-105)