In a recent survey, 76% of the community favored building a police substation in their neighborhood..?

If 14 citizens are chosen, find the probability that exactly 6 of them favor the building of the police substation?|||Let X be the number of people favoring the building. X has the binomial distribution with n = 14 trials and success probability p = 0.76





In general, if X has the binomial distribution with n trials and a success probability of p then


P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)


for values of x = 0, 1, 2, ..., n


P[X = x] = 0 for any other value of x.





The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.


Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.





X ~ Binomial( n = 14 , p = 0.76 )





the mean of the binomial distribution is n * p = 10.64


the variance of the binomial distribution is n * p * (1 - p) = 2.5536


the standard deviation is the square root of the variance = √ ( n * p * (1 - p)) = 1.597999





The Probability Mass Function, PMF,


f(X) = P(X = x) is:





P( X = 0 ) = 2.103572e-09


P( X = 1 ) = 9.325836e-08


P( X = 2 ) = 1.919568e-06


P( X = 3 ) = 2.431453e-05


P( X = 4 ) = 0.000211739


P( X = 5 ) = 0.001341014


P( X = 6 ) = 0.006369815 %26lt;%26lt;%26lt;%26lt;%26lt; ANSWER


P( X = 7 ) = 0.02305266


P( X = 8 ) = 0.06387509


P( X = 9 ) = 0.1348474


P( X = 10 ) = 0.2135084


P( X = 11 ) = 0.2458582


P( X = 12 ) = 0.1946377


P( X = 13 ) = 0.0948235


P( X = 14 ) = 0.02144817|||Use binomial distribution





14C6 (0.76)^6 (0.24)^8