Binomial Distribution how to do and.pptx

Binomial distribution

1. Binomial Distributions A coin-tossing experiment is a simple example of an important discrete random variable called the binomial random variable .

2. Example- A sales person interested in sale of a policy, if he visits 10 people in a day what is the probability of his selling policy to one person, or 2 person, ….

3. Definition: A binomial experiment is one that has these four characteristics: 1. The experiment consists of n identical results/ trials/ observation. 2. Each trial results in one of two outcomes: one outcome is called a success, S , and the other a failure, F. 3. The probability of success on a single trial is equal to p and remains the same from trial to trial. The probability of failure is equal to (1 - p ) = q. 4. The trials are independent.

4. The Binomial Probability function x = the number of successes p = the probability of a success on one trial n = the number of trials f ( x ) = the probability of x successes in n trials

5. Mean and Standard Deviation for the Binomial Random Variable: Mean: m = np Variance: s 2 = npq Standard deviation:

6. Binomial Formula.  Suppose a binomial experiment consists of  n  trials and results in  x  successes. If the probability of success on an individual trial is  P , then the binomial probability is: b( x ;  n, P ) =  n C x  * P x  * (1 - P) n – x b( x ;  n, P ) =  n C r  * P r  * (1 - P) n – r Where r is the value which the random variable takes

7. EXAMPLE Suppose a coin is tossed 2 times. What is the probability of getting 0 head Solution:  This is a binomial experiment in which the number of trials is equal to 2, the number of successes is equal to 0, and the probability of success on a single trial is 1/2. Therefore, the binomial probability is: b(0; 2, 0.5) =  2 C  * (0.5)  * (0.5) 2   = 1/4 (b) 1 head b(1; 2, 0.5) =  2 C 1  * (0.5) 1  * (0.5) 1   = 2/4 (c) 2 head b(2; 2, 0.5) =  2 C 2 * (0.5) 2  * (0.5)   = 1/4

8. EXAMPLE Suppose a coin is tossed 10 times. What is the probability of getting 0 head Solution:  This is a binomial experiment in which the number of trials is equal to 10, the number of successes is equal to 0, and the probability of success on a single trial is 1/2. Therefore, the binomial probability is: b(0; 10, 0.5) =  10 C  * (0.5)  * (0.5) 10   = 1/1024 Number of Trial 1 2 3 4 5 6 7 8 9 10 outcome T T T T T T T T T T Probability 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

9. EXAMPLE Suppose a coin is tossed 10 times. What is the probability of getting (b) 1 head Number of Trial 1 2 3 4 5 6 7 8 9 10 Outcome 1 H T T T T T T T T T

10. EXAMPLE Suppose a coin is tossed 10 times. What is the probability of getting (b) 1 head Number of Trial 1 2 3 4 5 6 7 8 9 10 Outcome 1 H T T T T T T T T T Outcome 2 T H T T T T T T T T

11. EXAMPLE Suppose a coin is tossed 10 times. What is the probability of getting (b) 1 head Number of Trial 1 2 3 4 5 6 7 8 9 10 Outcome 1 H T T T T T T T T T Outcome 2 T H T T T T T T T T Outcome 3 T T H T T T T T T T

12. EXAMPLE Suppose a coin is tossed 10 times. What is the probability of getting (b) 1 head Number of Trial 1 2 3 4 5 6 7 8 9 10 Outcome 1 H T T T T T T T T T Outcome 2 T H T T T T T T T T Outcome 3 T T H T T T T T T T Outcome 4 T T T H T T T T T T

13. EXAMPLE Suppose a coin is tossed 10 times. What is the probability of getting (b) 1 head Number of Trial 1 2 3 4 5 6 7 8 9 10 Outcome 1 H T T T T T T T T T Outcome 2 T H T T T T T T T T Outcome 3 T T H T T T T T T T Outcome 4 T T T H T T T T T T Outcome 5 T T T T H T T T T T Outcome 6 T T T T T H T T T T Outcome 7 T T T T T T H T T T Outcome 8 T T T T T T T H T T Outcome 9 T T T T T T T T H T Outcome 10 T T T T T T T T T H

14. EXAMPLE Suppose a coin is tossed 10 times. What is the probability of getting (b) 1 head Number of Trial 1 2 3 4 5 6 7 8 9 10 Outcome 1 H T T T T T T T T T Outcome 2 T H T T T T T T T T Outcome 3 T T H T T T T T T T Outcome 4 T T T H T T T T T T Outcome 5 T T T T H T T T T T Outcome 6 T T T T T H T T T T Outcome 7 T T T T T T H T T T Outcome 8 T T T T T T T H T T Outcome 9 T T T T T T T T H T Outcome 10 T T T T T T T T T H Probability 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

15. EXAMPLE Suppose a coin is tossed 10 times. What is the probability of getting (b) 1 head b(1; 10, 0.5) =  10 C 1   * (0.5) 1   * (0.5) 9   = 10/1024

16. EXAMPLE Suppose a coin is tossed 10 times. What is the probability of getting (c) 2 head b(2; 10, 0.5) =  10 C 2  * (0.5) 2  * (0.5) 8   = 45/1024

17. Approximation of Binomial to Normal

18. Approximation of Binomial to Normal

19. Approximation of Binomial to Normal

20. Approximation of Binomial to Normal

21. Approximation of Binomial to Normal

22. Approximation of Binomial to Normal

23. A manufacturer is making a product with a 20% defective rate. if we select 5 randomly chosen items at the end of the assembly line, what is the probability of having 1 defective items in our sample?

24. A survey conducted among Indian students found that 65% of them were very satisfied with their primary educational institution. Suppose a sample of 25 Indian students is taken. What is the probability that exactly 19 of them are very satisfied with their primary educational institution?

25. In a recent HR study, it was found that approximately 8% of job applicants for positions in a particular company do not meet the minimum qualifications for the job. If an HR manager randomly selects a sample of 30 job applicants , what is the probability that three or fewer of them do not meet the minimum qualifications?

26. As a Sales Manager you analyze the sales records for all sales persons under your guidance. Ram has a success rate of 75% and average 10 sales calls per day. Shyam has a success rate of 45% but average 16 calls per day What is the probability that each sales person makes 6 sales on any given day

27. In San Francisco, 30% of workers take public transportation daily In a sample of 10 workers, what is the probability that exactly three workers take public transportation daily In a sample of 10 workers, what is the probability that at least three workers take public transportation daily How many workers are expected to take public transportation daily? Compute the variance of the number of workers that will take the public transport daily. Compute the standard deviation of the number of workers that will take the public transportation daily.

28. In San Francisco, 30% of workers take public transportation daily In a sample of 10 workers, what is the probability that exactly three workers take public transportation daily= 0.2668 In a sample of 10 workers, what is the probability that at least three workers take public transportation daily= 0.6172 How many workers are expected to take public transportation daily? = 3 Compute the variance of the number of workers that will take the public transport daily. = 2.10 Compute the standard deviation of the number of workers that will take the public transportation daily.= 1.449

29. Twelve of the top twenty finishers in the 2009 PGA Championship at Hazeltine National Golf Club in Chaska, Minnesota, used a Titleist brand golf ball (Golf Ball Test website, November 12, 2009). Suppose these results are representative of the probability that a randomly selected PGA Tour player uses a Titleist brand golf ball. For a sample of 15 PGA Tour players, make the following calculation Compute the probability that exactly 10 of the 15 PGA Tour players use a Titleist brand golf ball Compute the probability that more than 10 of the 15 PGA Tour players use a Titleist brand golf ball For a sample of 15 PGA Tour players, compute the expected number of players who use a Titleist brand golf ball For a sample of 15 PGA Tour players, compute the variance and standard deviation of the number of players who use a Titleist brand golf ball

30. Using the 20 golfers in the Hazeltine PGA Championship, the probability that a PGA professional golfer uses a Titleist brand golf ball is p = 12/20 = .6 For the sample of 15 PGA Tour players, use a binomial distribution with n = 15 and p = .6 F(10)= 0.1859 P ( x >10) = f (11) + f (12) + f (13) + f (14) + f (15) .1268 + .0634 + .0219 + .0047 + .0005 = .2173 E ( x ) = np = 15(.6) = 9 Var ( x ) = s 2 = np (1 - p ) = 15(.6)(1 - .6) = 3.6 SD= 1.8974

31. A survey conducted by a financial research firm reports that 30% of individual investors in India have used a discount broker (a brokerage that does not charge full commissions). In a random sample of 10 individual investors , answer the following questions: What is the probability that exactly two of the sampled investors have used a discount broker? What is the probability that not more than three investors have used a discount broker? What is the probability that at least three of them have used a discount broker?

32. 12 out of 20 players in IPL used MRF Bat and won the match. For a randomly selected sample of 15 such players of IPL. Find the probability that exactly 10 out of 15 players use MRF bat. More then 10 players use MRF bat Find the expected number of players using MRF bat Find the SD and variance of the number of players using MRF bat