Probability question involving binomial distributions?

icy

May 2010
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Each question in a 15-question multiple-choice test had 5 possible answers. Suppose you guess randomly on each answer.
a)Show the probability distribution for the number of correct answers.(In a formula using x as the variable.)

b)Verify that the formula E(X)=np for the expectation of the number of correct answers.


Steps shown would be helpful.
 

Plato

MHF Helper
Aug 2006
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Each question in a 15-question multiple-choice test had 5 possible answers. Suppose you guess randomly on each answer.
a)Show the probability distribution for the number of correct answers.(In a formula using x as the variable.)
b)Verify that the formula E(X)=np for the expectation of the number of correct answers.
Steps shown would be helpful.
Showing some of your own work would be helpful also.
 
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Mar 2010
3
0
Each question in a 15-question multiple-choice test had 5 possible answers. Suppose you guess randomly on each answer.
a)Show the probability distribution for the number of correct answers.(In a formula using x as the variable.p=1/3,n=15,the rest you can do it

b)Verify that the formula E(X)=np for the expectation of the number of correct answers.
m(t)=(q+pexp^t)^n
m'(t)=npexp^t(q+pexp^t)^(n-1)
let t=0
E(X)=np(q+P)^(n-1)
=np(1-p+p)^(n-1) where q=1-p
=np(1)^(n-1)
=np
 

mr fantastic

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Dec 2007
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Each question in a 15-question multiple-choice test had 5 possible answers. Suppose you guess randomly on each answer.
a)Show the probability distribution for the number of correct answers.(In a formula using x as the variable.p=1/3,n=15,the rest you can do it

b)Verify that the formula E(X)=np for the expectation of the number of correct answers.
m(t)=(q+pexp^t)^n
m'(t)=npexp^t(q+pexp^t)^(n-1)
let t=0
E(X)=np(q+P)^(n-1)
=np(1-p+p)^(n-1) where q=1-p
=np(1)^(n-1)
=np
I suspect that part (b) of the question requires the OP to explicitly calculate the mean using the answer to part (a), and then show that it gives the same value as the formula np.