Need help with Statistics homework. I need someone to walk me through the formula. Thanks!?

6. In a large university, 15% of the students are female. If a random sample of twenty students is selected,





a. What is the probability that the sample contains exactly 4 female students?





b. What is the probability that the sample will contain no female students?





c. What is the probability that the sample will contain exactly 20 female students?





d. What is the probability that the sample will contain no more than 9 female students?





e. What is the probability that the sample will contain fewer than 5 female students?





f. What is the expected number of female students?

Need help with Statistics homework. I need someone to walk me through the formula. Thanks!?
Let X be the number of female students in the sample of twenty. X has the binomial distribution with n = 20 trials and success probability p = 0.15.





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.





this is found 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, to be more accurate, the binomial is the sum of n independent and identically distributed Bernoulli trials.





X ~ Binomial( n , p )





the mean of the binomial distribution is n * p


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


the standard deviation is the square root of the variance.





P(X = 0 ) = 0.03875953 %26lt;= answer to B


P(X = 1 ) = 0.1367983


P(X = 2 ) = 0.2293384


P(X = 3 ) = 0.2428289


P(X = 4 ) = 0.1821217 %26lt;= answer to A


P(X = 5 ) = 0.1028452


P(X = 6 ) = 0.04537287


P(X = 7 ) = 0.01601396


P(X = 8 ) = 0.004592237


P(X = 9 ) = 0.001080526


P(X = 10 ) = 0.0002097492


P(X = 11 ) = 3.364961e-05


P(X = 12 ) = 4.453625e-06


P(X = 13 ) = 4.836516e-07


P(X = 14 ) = 4.267514e-08


P(X = 15 ) = 3.012363e-09


P(X = 16 ) = 1.661229e-10


P(X = 17 ) = 6.897839e-12


P(X = 18 ) = 2.028776e-13


P(X = 19 ) = 3.768624e-15


P(X = 20 ) = 3.325257e-17 %26lt;= Answer to C





P(X ≤ 0 ) = 0.03875953


P(X ≤ 1 ) = 0.1755579 = P(X = 0) + P(X = 1)


P(X ≤ 2 ) = 0.4048963 = P(X = 0) + P(X = 1) + P(X = 2)


P(X ≤ 3 ) = 0.6477252


P(X ≤ 4 ) = 0.8298468


P(X ≤ 5 ) = 0.932692 %26lt;= answer to e


P(X ≤ 6 ) = 0.9780649


P(X ≤ 7 ) = 0.9940789


P(X ≤ 8 ) = 0.998671


P(X ≤ 9 ) = 0.9997516 %26lt;= Answer to D


P(X ≤ 10 ) = 0.9999614


P(X ≤ 11 ) = 0.999995


P(X ≤ 12 ) = 0.9999995





expected value is n * p = 20 * 0.15 = 3