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Binomial distribution Nutan S. Mishra Department of Mathematics and Statistics University of South Alabama
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Binomial experiment An experiment is called binomial experiment if it satisfies following four conditions 1.Consists of n trials 2.All the trials are independent. 3.There are only two possible outcomes of each trial. 4.Probability of success in each trial is constant say p. X is number of successes in the experiment
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Example Toss four fair coins. X= # heads showed up Then x may take values 0 or 1 or 2 or 3 or at the most 4. p =.5, n =4. Use binomial table to complete the following table xP(X) = 4 C x p x (1-p) 4-x 0 4 C 0 p 0 (1-p) 4 = 1 4 C 1 p 1 (1-p) 3 = 2 4 C 2 p 2 (1-p) 2 = 3 4 C 3 p 3 (1-p) 1 = 4 4 C 4 p 4 (1-p) 0 =
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Example In a company it is known that among the population of employees 35% are smokers and 65% are non smokers. If we select a sample of size 10 employees from this population and count the number of smokers is the sample then X= # smokers in the sample of 10 is a binomial variable with n=10 and p=.35 that is (1-p) =.65. The possible values x takes : 0 to 10 Use table of combinations and a calculator to complete the table on next slide
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X # smokers in the sample of size10 P(X)= 10 C x p x (1-p) 10-x 0 10 C 0 p 0 (1-p) 10 = 1 10 C 1 p 1 (1-p) 9 = 2 10 C 2 p 2 (1-p) 8 = 3 10 C 3 p 3 (1-p) 7 = 4 10 C 4 p 4 (1-p) 6 = 5 10 C 5 p 5 (1-p) 5 = 6 10 C 6 p 6 (1-p) 4 = 7 10 C 7 p 7 (1-p) 3 = 8 10 C 8 p 8 (1-p) 2 = 9 10 C 9 p 9 (1-p) 1 = 10 10 C 10 p 10 (1-p) 0 =
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