# Probability question involving binomial distributions?

#### icy

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.

#### Plato

MHF Helper
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.

• mr fantastic

#### edison

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

MHF Hall of Fame
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.