Question

I need to use factorials and doing it step-by-step for the bio normal probability formula Please help me figure this A through C step-by-step they want the factorial symbol! When showing my workGiven the number of trials and the probability of success, determine the probability indicated: (Hint use binomial distribution formula use factorials ! in showing your work)n = 15, p = 0.4, find P(4 successes) n = 12, p = 0.2, find P(2 success ) n = 20, p = 0.05, find P(at most 3 successes) (hint for c. P (at most 3 successes) = P(x ≤3)= P(x= 0) + P(x = 1)+ P(x = 2)+ P(x = 3)I just got disconnected from my tutor who said it would take 20 minutes to song and help me go over it step-by-step so if you were that tutor or any tour available please contact

Answer 1

Binomial distribution formula:

$P(x)=\frac{n!}{(n-x)!x!}*p^x*q^{n-x}$

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n = 15, p = 0.4, find P(4 successes)

$\begin{gathered} n=15 \\ x=4 \\ p=0.4 \\ q=1-0.4=0.6 \\ \\ P(x=4)=\frac{15!}{(15-4)!*4!}*0.4^4*0.6^{15-4} \\ \\ P(x=4)=\frac{15!}{11!*4!}*0.4^4*0.6^{11} \\ \\ P(x=4)=\frac{15\times14\times13\times12\times11!}{11!*4!}*0.4^4*0.6^{11} \\ \\ P(x=4)=\frac{15\times14\times13\times12}{4\times3\times2\times1}*0.4^4*0.6^{11} \\ \\ P(x=4)=1365*0.4^4*0.6^{11} \\ \\ P(x=4)=0.1268 \end{gathered}$

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n = 12, p = 0.2, find P(2 success )

$\begin{gathered} n=12 \\ x=2 \\ p=0.2 \\ q=1-0.2=0.8 \\ \\ P(x=2)=\frac{12!}{(12-2)!*2!}*0.2^2*0.8^{12-2} \\ \\ P(x=2)=\frac{12!}{10!*2!}*0.2^2*0.8^{10} \\ \\ P(x=2)=\frac{12\times11\times10!}{10!*2!}*0.2^2*0.8^{10} \\ \\ P(x=2)=\frac{12\times11}{2\times1}*0.2^2*0.8^{10} \\ \\ P(x=2)=66*0.2^2*0.8^{10} \\ \\ P(x=2)=0.2835 \end{gathered}$

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n = 20, p = 0.05, find P(at most 3 successes)

Find each part (P(x=0), P(x=1), P(x=2), P(x=3)) and then sum the results

$\begin{gathered} n=20 \\ p=0.05 \\ q=1-0.05=0.95 \\ \\ \end{gathered}$$\begin{gathered} P(x=0)=\frac{20!}{(20-0)!0!}*0.05^0*0.95^{20-0} \\ \\ P(x=0)=\frac{20!}{20!*0!}*1*0.95^{20} \\ \\ P(x=0)=\frac{20!}{20!*1}*1*0.95^{20} \\ P(x=0)=1*1*0.95^{20} \\ P(x=0)=0.3585 \end{gathered}$$\begin{gathered} P(x=1)=\frac{20!}{19!*1!}*0.05^1*0.95^{19} \\ \\ P(x=1)=\frac{20\times19!}{19!*1}*0.05*0.95^{19} \\ \\ P(x=1)=20*0.05*0.95^{19} \\ P(x=1)=0.3774 \\ \end{gathered}$$\begin{gathered} P(x=2)=\frac{20!}{18!*2!}*0.05^2*0.95^{18} \\ \\ P(x=2)=\frac{20\times19\times18!}{18!*2\times1}*0.05^2*0.95^{18} \\ \\ P(x=2)=190*0.05^2*0.95^{18} \\ P(x=2)=0.1887 \end{gathered}$$\begin{gathered} P(x=3)=\frac{20!}{17!*3!}*0.05^3*0.95^{17} \\ \\ P(x=3)=\frac{20\times19\times18\times17!}{17!*3\times2\times1}*0.05^3*0.95^{17} \\ \\ P(x=3)=1140*0.05^3*0.95^{17} \\ P(x=3)=0.0596 \end{gathered}$$P(x\leq3)=0.3585+0.3774+0.1887+0.0596=0.9842$