Question

find the probability of exactly three
successes in eight trials of a binomial
experiment in which the probability of
success is 45%.
n = 8 p = 0.45 q = 0.55 k = 3
enter values into the formula.
p(k) = _nc_k(p)^k(q)^{n - k}
p(?) = _c_ ( ) ( )^{ - }

Explanation:

Step1: Substitute values into formula

The binomial probability formula is \(P(k)={}_nC_k(p)^k(q)^{n - k}\). Here \(n = 8\), \(k=3\), \(p = 0.45\), \(q=0.55\). So \(P(3)={}_8C_3(0.45)^3(0.55)^{8 - 3}\).

Step2: Calculate combination

The combination formula is \({}_nC_k=\frac{n!}{k!(n - k)!}\). For \(n = 8\) and \(k = 3\), \({}_8C_3=\frac{8!}{3!(8 - 3)!}=\frac{8!}{3!5!}=\frac{8\times7\times6}{3\times2\times1}=56\).

Step3: Calculate powers

\((0.45)^3=0.45\times0.45\times0.45 = 0.091125\) and \((0.55)^{5}=0.55\times0.55\times0.55\times0.55\times0.55\approx0.0503284375\).

Step4: Multiply values

\(P(3)=56\times0.091125\times0.0503284375\approx0.256\).

Answer:

\(P(3)={}_8C_3(0.45)^3(0.55)^{5}\approx0.256\)