Notes On Binomial Distribution - CBSE Class 12 Maths
For n Bernoulli trials:     X       0       1       2     ...       n   P(X) 1st term of (q + p)n 2nd term of (q + p)n 3rd term of (q + p)n ... (n + 1)th term of (q + p)n      X         0         1          2 ...           n     P(X) nC0qn-0p0 nC1qn-1p1 nC2qn-2p2 ... nCnqn-npn This probability distribution of a random variable, x, is called a binomial distribution. Thus, the general formula for the probability of a given number of successes is given by… P(X = x) = P(x) = nCxqn-xpx             = n!/(x!(n-x)!) qn-xpx, where x = 0,1,2,....,n and q = 1 - p P (x) is known as the probability function of the binomial distribution. Ex: Find the probability of getting two heads, if a coin is tossed six times. Sol: Let the event of getting a head on tossing a coin be a success, and the event of getting a tail be a failure. Let X denote the number of heads obtained, when a coin is tossed six times. Let p be the probability of getting a head and q be the probability of getting a tail, if a coin is tossed once. ∴ p = 1/2 and q = 1/2 Number of trials, n = 6 ∴ X has a binomial distribution with n =6 and p = 1/2 We have P(X = x) = nCxqn-xpx , where x = 0,1,2,3,....,n The probability of getting 2 heads = P(X = 2) = 6C2(1/2)6-2(1/2)2 = 6C2(1/2)6 = 6!/(2!4!) (1/2)6 = (6x5)/(2x1)  . (1/2)6 = 15/64 Hence, the probability of getting 2 heads, when a coin is tossed six times = 15/64

#### Summary

For n Bernoulli trials:     X       0       1       2     ...       n   P(X) 1st term of (q + p)n 2nd term of (q + p)n 3rd term of (q + p)n ... (n + 1)th term of (q + p)n      X         0         1          2 ...           n     P(X) nC0qn-0p0 nC1qn-1p1 nC2qn-2p2 ... nCnqn-npn This probability distribution of a random variable, x, is called a binomial distribution. Thus, the general formula for the probability of a given number of successes is given by… P(X = x) = P(x) = nCxqn-xpx             = n!/(x!(n-x)!) qn-xpx, where x = 0,1,2,....,n and q = 1 - p P (x) is known as the probability function of the binomial distribution. Ex: Find the probability of getting two heads, if a coin is tossed six times. Sol: Let the event of getting a head on tossing a coin be a success, and the event of getting a tail be a failure. Let X denote the number of heads obtained, when a coin is tossed six times. Let p be the probability of getting a head and q be the probability of getting a tail, if a coin is tossed once. ∴ p = 1/2 and q = 1/2 Number of trials, n = 6 ∴ X has a binomial distribution with n =6 and p = 1/2 We have P(X = x) = nCxqn-xpx , where x = 0,1,2,3,....,n The probability of getting 2 heads = P(X = 2) = 6C2(1/2)6-2(1/2)2 = 6C2(1/2)6 = 6!/(2!4!) (1/2)6 = (6x5)/(2x1)  . (1/2)6 = 15/64 Hence, the probability of getting 2 heads, when a coin is tossed six times = 15/64

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