Binomial Proportions

Question

On a multiple choice test with ten questions with five choices for each question if a student guesses at every answer find the probability of the following....

 

A) more than 4 correct answers

 

B) atleast 1 correct answer

 

C) exactly 3 correct answers

Answer 1

I use Excel here, but if you have a calculator it should have similar functions.

 

With the binomial distribution, P(x) = C(n,x)p^x(1-p)^(n-x)

With 5 choices, p = 1/5 = .2; 1-p = .8; 10 questions means n = 10; x is the number of correct answers

We can get the answer to this question by using this formula or using Excel's binom.dist function.

 

a)P(more than 4 correct answers) = 1 - P(4 or less correct answers) 

1 - ( C(10,0).2^0(.8)^(10-0) + C(10,1).2^1(.8)^(9) + C(10,2).2^2(.8)^(10-2) + C(10,3).2^3(.8)^(7) + C(10,4).2^4(.8)^(6)) =  in Excel,

=1 - ( COMBIN(10,0)*0.2^0*(0.8)^(10-0) + COMBIN(10,1)*0.2^1*(0.8)^(9) + COMBIN(10,2)*0.2^2*(0.8)^(10-2) + COMBIN(10,3)*0.2^3*(0.8)^(7) + COMBIN(10,4)*0.2^4*(0.8)^(6)) = 0.032793498

 

A shorcut in Excel is to use the binom.dist function.

1 - P(X <= 4) = 1-BINOM.DIST(4,10,0.2,1) =  0.032793498 (the final 1 is for cumulative) - the same answer as above

 

b) at least 1 correct answer = 1 - P(0 correct answers) = 1 - .8^10 = 0.892625818

 

c)P(3) = C(10,3) .2^3 * .8^7 = 120 * .008 * 0.2097152 = 0.201326592

Alternatively, binom.dist(3,10,.2,0) = 0.201326592   (note the final 0 rather than 1 because we are using P(x = 3) rather than P(x <= 3)