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Statistics 211/Labpack/Binomial Distribution Binomial: Goals of the lab: To give students some familiarity with the Binomial distribution. Background: The binomial distribution is a fairly common discrete random variable. There are four conditions that must hold for a RV to be a binomial random variable. • There must be a fixed number of trials, n. • Each trial can result in one and only one of two possible outcomes, labeled “success” and “failure”. • The probability of success in a single trial, p, is the same for each trial. • The trials are independent, so that the probability of success is unaffected by the result of a previous trial. The probability distribution function is given below*. The textbook we are using this semester has a table that requires some elaboration. Table G in our textbook "Statistics The Craft of Data Collection, Description, and Inference (by Monrad, et.al.) is a cumulative binomial table. This means that the values in the table are P(Y£ r) for some r (and, of course, n and p). Consequently if we want to calculate other binomial probabilities, then we must do some calculations. The table below illustrates those rules. Relevant Formulae: * P(X Ê nˆ x n-x = x) = Á p (1 - p) for x=0,1,2,…,n Ë x¯ Rules for Table G in textbook: Probability we want P(Y £ r) P(Y < r) P(Y ≥ r) P(Y > r) *P(Y=r) Calculation we need to perform Example P(Y£ r) P(Y£(r-1)) 1-P(Y£(r-1)) 1-P(Y£r) P(Y£r)-P(Y£(r-1)) P(Y£4) P(Y<4) = P(Y£3) P(Y≥4) = 1-P(Y£3) P(Y>4) = 1-P(Y£4) P(Y=4) = P(Y£4)-P(Y£3) Examples: There are four conditions that must hold for a RV to be a binomial random variable. • There must be a fixed number of trials, n. • Each trial can result in one and only one of two possible outcomes, labeled “success” and “failure”. • The probability of success in a single trial, p, is the same for each trial. • The trials are independent, so that the probability of success is unaffected by the result of a previous trial. Suppose that n = 20, p = 0.35, and X = the number of successes. P(X=6) = ÊÁ nˆ x Ê 20ˆ p (1 - p)n- x = Á (0.35)6 (1 - 0.35)20-6 Ë x¯ Ë 6¯ Ê 20ˆ 0.3560.6514 Ë 6¯ =Á = (38760)*(0.001838)*(0.002403) = 0.1712 We can also use Table G in textbook to calculate this: P(X=6) = P(X£6) – P(X£5) = 0.4166 – 0.2454 = 0.1712 Other examples that use the table of cumulative binomial probabilities: P(X>3) = P(X≥4) = 1- P(X£ 3) = 1 – 0.0444 = 0.9556 P(X<10) = P(X£9) = 0.8782 Page 1 of 1