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

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

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

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

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

9. Example: Tossing a Coin 5 times xf(x) 00.03125 10.15625 20.3125 3 40.15625 50.03125 n = 5 x = # heads in 5 tosses p = P(head) = 0.5

10. Binomial Distribution n 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 n 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. Tree Diagram Binomial Distribution 1 st Worker 2 nd Worker 3 rd Worker x x Prob. Leaves (.1) Leaves (.1) Stays (.9) Stays (.9) 3 3 2 2 0 0 2 2 2 2 Leaves (.1) S (.9) Stays (.9) S (.9) L (.1).0010.0090.7290.0090 1 1 1 1.0810 11

13. n Using Tables of Binomial Probabilities Binomial Distribution

14. E ( x ) =  = np Var( x ) =  2 = np (1  p ) Expected Value Variance Standard Deviation

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

16. 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 Example

17. 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