Binomial Distribution (Dr. Monticino)

Assignment Sheet  Read Chapter 15  Assignment # 9 (Due March 30 th )  Chapter 15  Exercise Set A: 1-6  Review exercises: 1,2,3,4 (important),5,6,7,10,11  Re-do example problems in last two lectures  Exam 2 is projected to be on April 11 th or 13 th depending on when we finish Chapter 18

Overview  Binomial Model  Assumptions  Calculating Probabilities  Examples  Law of Averages

Binomial Model  The binomial distribution is used as a model for a process which is repeated n times.  Each time the process is repeated, outcomes are classified as either successes or failures  Each time the process is repeated there is the same probability of a success occurring  Successive outcomes are independent of one another

Binomial Distribution  Under the assumptions of the binomial model, the probability of k successes out of n repetitions is

Examples  Flip a fair coin 10 times  What is the probability that 10 heads come up?  What is the probability that exactly 8 tails occur?  What is the probability that at least 8 tails occur?

Examples  Roll two fair die 5 times  What is the probability that 5 “doubles” are rolled?  What is the probability that doubles are rolled at most twice  What is the probability that the sum of the die is seven on 3 out of the 5 rolls

Examples  The likelihood of a women developing breast cancer during her lifetime is 1 in 9  Suppose 8 women are randomly chosen from the population  What is the probability that they all develop breast cancer  What is the probability that at least two will develop breast cancer?

Law of Averages  The law of averages says that if a chance process is repeated a large number of times, then the percentage of times that a particular event occurs is likely to be close to the probability of that event  Provided the assumptions assumed for the binomial model hold  There is always chance error (Dr. Monticino)