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)=(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)=(15!)/((15-4)!*4!)*0.4^4*0.6^(15-4) \\ \\ P(x=4)=(15!)/(11!*4!)*0.4^4*0.6^(11) \\ \\ P(x=4)=(15*14*13*12*11!)/(11!*4!)*0.4^4*0.6^(11) \\ \\ P(x=4)=(15*14*13*12)/(4*3*2*1)*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)=(12!)/((12-2)!*2!)*0.2^2*0.8^(12-2) \\ \\ P(x=2)=(12!)/(10!*2!)*0.2^2*0.8^(10) \\ \\ P(x=2)=(12*11*10!)/(10!*2!)*0.2^2*0.8^(10) \\ \\ P(x=2)=(12*11)/(2*1)*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)=(20!)/((20-0)!0!)*0.05^0*0.95^(20-0) \\ \\ P(x=0)=(20!)/(20!*0!)*1*0.95^(20) \\ \\ P(x=0)=(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)=(20!)/(19!*1!)*0.05^1*0.95^(19) \\ \\ P(x=1)=(20*19!)/(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)=(20!)/(18!*2!)*0.05^2*0.95^(18) \\ \\ P(x=2)=(20*19*18!)/(18!*2*1)*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)=(20!)/(17!*3!)*0.05^3*0.95^(17) \\ \\ P(x=3)=(20*19*18*17!)/(17!*3*2*1)*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`