Question

I do not understand the factorial! I was told I need to have that in all my work with bionomal probability formula

Answer 1

In order to calculate the probability of k successes among n trials, we can use the formula below:

$P(x=k)=C(n,k)\cdot p^k\cdot(1-p)^{n-k}$

Where C(n, k) is the combination of n choose k:

$C(n,k)=\frac{n!}{k!(n-k)!}$

For n = 15, p = 0.4 and k = 4, we have:

$\begin{gathered} P(x=4)=C(15,4)\cdot0.4^4\cdot0.6^{11}\\ \\ P(x=4)=\frac{15!}{4!11!}\cdot0.4^4\cdot0.6^{11}\\ \\ P(x=4)=\frac{15\cdot14\cdot13\cdot12}{4\cdot3\cdot2}\cdot0.4^4\cdot0.6^{11}\\ \\ P(x=4)=1365\cdot0.4^4\cdot0.6^{11}\\ \\ P(x=4)=0.12677 \end{gathered}$

For n = 12, p = 0.2 and k = 2, we have:

$\begin{gathered} P(x=2)=C(12,2)\cdot0.2^2\cdot0.8^{10}\\ \\ P(x=2)=0.28347 \end{gathered}$

For n = 20, p = 0.05 and k = 0, 1, 2 and 3, we have:

$\begin{gathered} P(x=0)=C(20,0)\cdot0.05^0\cdot0.95^{20}=0.358486\\ \\ P(x=1)=C(20,1)\cdot0.05^1\cdot0.95^{19}=0.377354\\ \\ P(x=2)=C(20,2)\cdot0.05^2\cdot0.95^{18}=0.188677\\ \\ P(x=3)=C(20,3)\cdot0.05^3\cdot0.95^{17}=0.059582\\ \\ \\ \\ P(x\leq3)=P(x=0)+P(x=1)+P(x=2)+P(x=3)=0.9841=98.41\% \end{gathered}$