Business and Finance College Principles of Statistics Eng. Heba Hamad 2008

1. Business and Finance College Principles of StatisticsEng. Heba Hamad2008

2. Slides Prepared by JOHN S. LOUCKS St. Edward’s University

3. Binomial Distribution Four Properties of a Binomial Experiment • 1. The experiment consists of a sequence of n • identical trials. • 2. Two outcomes, success and failure, are possible • on each trial. 3. The probability of a success, denoted by p, does not change from trial to trial. 4. The trials are independent.

4. Binomial Distribution Our interest is in the number of successes occurring in the n trials. We let x denote the number of successes occurring in the n trials.

5. Example of Tossing a Coin • Toss a coin 5 times in succession • Is this experiment binomial? • What is success? • What is n? • What is x?

6. Example of Tossing a Coin • Toss a coin 5 times in succession • Is this experiment binomial? Yes • What is success? Let’s define it as “heads” • What is n? 5 • What is x? Can take on the values of 0, 1, 2, 3, 4, 5 – depending on the number of “heads” obtained

7. Binomial Distribution • Binomial Probability Function where: f(x) = the probability of x successes in n trials n = the number of trials p = the probability of success on any one trial

8. Binomial Distribution • Binomial Probability Function Probability of a particular sequence of trial outcomes with x successes in n trials Number of experimental outcomes providing exactly x successes in n trials

9. Example: Tossing a Coin 5 times n = 5 x = # heads in 5 tosses p = P(head) = 0.5

10. Binomial Distribution • Example: Evans Electronics Evans is concerned about a low retention rate for employees. In recent years, management has seen a turnover of 10% of the hourly employees annually. Thus, for any hourly employee chosen at random, management estimates a probability of 0.1 that the person will not be with the company next year.

11. Binomial Distribution • Using the Binomial Probability Function Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year? Let: p = .10, n = 3, x = 1

12. Binomial Distribution • Tree Diagram x 1st Worker 2nd Worker 3rd Worker Prob. L (.1) .0010 3 Leaves (.1) .0090 2 S (.9) Leaves (.1) L (.1) .0090 2 Stays (.9) 1 .0810 S (.9) L (.1) 2 .0090 Leaves (.1) Stays (.9) 1 .0810 S (.9) L (.1) 1 .0810 Stays (.9) 0 .7290 S (.9)

13. Binomial Distribution • Using Tables of Binomial Probabilities

14. E(x) =  = np • Var(x) =  2 = np(1 -p) Binomial Distribution • Expected Value • Variance • Standard Deviation

15. E(x) =  = 3(.1) = .3 employees out of 3 Var(x) =  2 = 3(.1)(.9) = .27 Binomial Distribution • Expected Value • Variance • Standard Deviation

16. Example Use the binomial probability formula to find the probability of getting exactly 3 correct responses among 5 different requests from directory assistance. Assume that in general the responses is correct 90% of the time. That is Find P(3) given that n=5, x=3, p=0.9 & q=0.1

17. Example Consider the experiment of flipping a coin 3 times. If we let the event of getting tails on a flip be considered “success”, and if the random variable T represents the number of tails obtained, then T will be binomially distributed with n=3 ,p=0.5 , and q=0.5 . calculate the probability of exactly 2 tails