LESSON 9: BINOMIAL DISTRIBUTION

Transcription

1 LESSON 9: Outline The context The properties Notation Formula Use of table Use of Excel Mean and variance 1 THE CONTEXT An important property of the binomial distribution: An outcome of an experiment is classified into one of two mutually exclusive categories - success or failure. Example: Suppose that a production lot contains 100 items. The producer and a buyer agree that if at most 2 out of a sample of 10 items are defective, then all the remaining 90 items in the production lot will be purchased without further testing. Note that each item can be defective or non defective which are two mutually exclusive outcomes of testing. Given the probability that an item is defective, what is the probability that the 90 items will be purchased without further testing? 2 1

2 THE CONTEXT Trial Flip a coin Apply for a job Answer a Multiple choice question Two Mut. Excl. and exhaustive outcomes Head / Tail Get the job / not get the job Correct / Incorrect 3 THE ROERTIES The binomial distribution has the following properties: 1. The experiment consists of a finite number of trials. The number of trials is denoted by n. 2. An outcome of an experiment is classified into one of two mutually exclusive categories - success or failure. 3. The probability of success stays the same for each trial. The probability of success is denoted by p. 4. The trials are independent. 4 2

3 THE NOTATION Notation n : the number of trials r : the number of observed successes p : the probability of success on each trial Note: n-r : the number of observed failures 1- p : the probability of failure on each trial 5 THE ROBABILITY DISTRIBUTION The binomial probability distribution gives the probability of getting exactly r successes out of a total of n trials. The probability of getting exactly r successes out of a total of n trials is as follows: b r; n, π = R = r = ( ) ( ) n r n r C π ( 1 π ) r n! ( ) r ( ) n = π 1 π r r! n r! n Note: In the above C r gives the number of different ways of choosing r objects out of a total of n objects 6 3

4 THE ROBABILITY DISTRIBUTION Example 1: If you toss a fair coin twice, what is the probability of getting one head and one tail? Use the binomial probability distribution formula. 7 THE ROBABILITY DISTRIBUTION Example 2: Redo Example 1 with a probability tree and verify if the probability tree gives the same answer. 8 4

5 THE CUMULATIVE ROBABILITY The cumulative probability gives the probability of getting at most r successes out of a total of n trials. The probability of getting at most r successes out of a total of n trials is as follows: B = ( r; n, π ) = ( R r) r x= 0 ( x; n, π ) Note: An uppercase B(r) is used to distinguish the cumulative probability distribution function from the probability mass function b(r) b 9 THE CUMULATIVE ROBABILITY Example 3: If you toss a fair coin three times, what is the probability of getting at most one head (at least two tails)? 10 5

6 THE CUMULATIVE ROBABILITY Example 4: Redo Example 3 with a probability tree and verify if the probability tree gives the same answer. 11 NECESSITY OF A TABLE OR SOFTWARE Example 5 (do not solve): If you toss a fair coin 50 times, what is the probability of getting at most 20 heads (at least 30 tails)? Do not solve this problem, but discuss the computation required by the binomial probability distribution formula. 12 6

7 USE OF TABLE B( r;n,π ) ( R r) Table A, Appendix A, pp gives the probability of getting at most r successes out of a total of n trials, for probability of success in each trial p. The table can be used to find the probability of ( ) ( ) ( ( 1) ) exactly r successes: R = r = R r R r at least r successes: ( R r) = 1 ( R ( r 1) ) successes between a and b: ( a R b) = ( R r ) ( R ( r 1) ) 13 Example 6: Find the following using Table A: ( R 1 n = 5, π = 0.30) ( R 2 n = 5, π = 0.30) ( R 4 n = 10, π = 0.30) Example 7: Find the following using above values ( R = 2 n = 5, π = 0.30) ( R 3 n = 5, π = 0.30) ( 2 R 4 n = 5, π = 0.30) USE OF TABLE 14 7

8 USE OF TABLE Example 8: If you toss a fair coin 50 times, what is the probability of getting at most 20 heads (at least 30 tails)? Solve this problem using the Table A. 15 USE OF EXCEL ( ) ( ) The Excel function BINOMDIST gives R r and R = r It takes four arguments. The first 3 arguments are r,n,p The last one is TRUE for ( R r) and FALSE for R = r ( ) Example 9: If you toss a fair coin 50 times, what is the probability of getting at most 20 heads (at least 30 tails)? Solve this problem using Excel. Verify if Excel gives the same answer as it is given by Table A in Example 8. Answer: ( R 20 n = 50, π = 0.50) =BINOMDIST(20,50,0.5,TRUE) 16 8

9 MEAN AND VARIANCE The expected value and variance for the number of successes R may be computed as follows: E ( R) = nπ ( R) = nπ ( 1 π ) Var E(R) is the mean or expected value of R Var(X) is the variance of R n is the number of trials p is the probability of success on each trial The probability of failure on each trial = 1- p 17 MEAN AND VARIANCE Example 10: Let R be a random variable that gives number of heads when a fair coin is tossed 4 times. Compute E(R) and Var(R). 18 9

10 READING AND EXERCISES Lesson 9 Reading: Section 7-3, pp Exercises: 7-22, 7-24,