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• Binomial Distribution Conditions for Binomial Distribution • Each trial has two possible outcomes (“success” and “failure”). • Each trial has the same probability of a success, denoted by p. • The n trials are independent. The binomial random variable X is the nnumber of successes in the n trials. Probabilities for a Binomial Distribution For n independent trials, the probability of x successes equals P(x) = n! p x (1 − p)n−x , x!(n − x)! x = 0, 1, 2, . . . , n Here p is the probability for a success of each trial. Guideline for Using Binomial Distribution For sampling n subjects from a population without replacement, if n is less than 10% of the population size, then we can use the the binomial distribution to approximate the exact probability distribution of the number of successes. (In other words, we pretend it is sampling WITH replacement.) Mean and Standard Deviation The mean µ of the binomial distribution for n trials with “success” probability p is µ = np And the standard deviation is σ= p np(1 − p) Approximate The Binomial Distribution by The Normal Distribution The binomial distribution can be approximated well by the normal distribution when (i) n is large enough (≤ 40); (ii) the expected number of successes np is at least 15; and (iii) the expected number of failures np is at least 15.