2. Binomial Distribution If you flip a coin 3 times, what is the probability that you will get exactly 1 tails? There is more than one way to do this problem, but the easiest is by using the methods for a binomial distribution.

3. Binomial Distribution Binomial distribution applies to cases where there are only two possible outcomes: head or tail, success or failure and defective item or good item. Requirements for using Binomial Distribution -Each outcome is independent of the others -There are only two possible outcomes (success or failure) -The probability of each outcome is always the same -The number of trials must be fixed.

4. Here are some exercises to look at. Determine if X has a binomial distribution, and why? 1) You observe the sex of the next 20 children born at a local hospital; X is the number of girls among them. 2) A couple decides to continue to have children until their first girl is born; X is the total number of children the couple has. 3) Joe buys a state lottery ticket every week. The count X is the number of times a year that he wins a prize.

5. 1) It may be binomial if we assume that there are no twins or other multiple births among the ext 20, and that for all births, the probability that the baby is female is the same. 2) No – the number of observations is not fixed. 3) Assuming that Joe’s chance of winning the lottery is the same every week and that a year consists of 52 weeks (observations), this would be binomial.

6. Binomial Distribution Let p = probability that an item is defective q = probability that an item is good The item can either be good or defective. Therefore, q = 1-p If n items are produced, find the probability of exactly r defective items.

7. Binomial Distribution The formula for finding the probability of “r” things occurring out of “n” trials is… Where p is the probability of a success and q is the probability of a failure and…

8. Back to the coin problem For our problem p=probability of a tails (.5) and q=probability of a heads (.5) so our equation would be…

9. Binomial Distribution A sample of eleven electric bulbs is drawn every day from those manufactured at a plant. Probabilities of defective bulbs are random and independent of previous results. The probability that a bulb is defective is 0.04. What is the probability of finding exactly three defective bulbs in a sample? What is the probability of finding three or more defective bulbs in a sample?

10. Binomial Distribution No. of defective bulbs Probability 0(1)(0.04) 0 (0.96) 11 = 0.6382 1 11 C 1 (0.04) 1 (0.96) 10 = 0.2925 2 11 C 2 (0.04) 2 (0.96) 9 = 0.0609 3 11 C 3 (0.04) 3 (0.96) 8 = 0.0076

11. Binomial Distribution Pr[3 defective bulbs] = 0.0076 Pr[3 or more def.] = 1- Pr[0 def.] – Pr[1 def.] – Pr[2 def.] = 1 – 0.6382 - 0.2925 – 0.0609 = 0.0084

12. Binomial Distribution Standard Deviation Mean Variance

13. Binomial Distribution For p = 0.35 and n = 10