Question

A student is taking a​ multiple-choice exam in which each question has five choices. Assuming that she has no knowledge of the correct answers to any of the​ questions, she has decided on a strategy in which she will place five balls​ (marked Upper A comma Upper B comma Upper C comma Upper D comma and Upper E​) into a box. She randomly selects one ball for each question and replaces the ball in the box. The marking on the ball will determine her answer to the question. There are four ​multiple-choice questions on the exam.

What is the probability that she will get no questions​ correct?

Please include excel formula.

Answer 1

This is the case of Binomial Distribution

Probability mass function for Binomial distribution is :

P(X=x) = ⁿC_x p^x (1-p)^(n-x)

where,

p = Probability of success

n = Number of trials

x = Number of success

ⁿC_x = n! / x! * ( n-x )!

n! = n*(n-1)*(n-2)* ............... * 1

Now, in this case we have,

Since, there are 5 choices for a question,

So, Probability of randomly getting a question correct = 1/5

p = Probability of getting a question correct = 1/5

n = Number of questions = 4

x = Number of correct answers = 0

So, Probability that she will get no questions​ correct =

P(X=0) = ⁴C₀ (1/5)⁰ (1-1/5)^{(4 - 0)}

⁴C₀ = 4! / ( 0! * (4-0)! ) = 4 ! / 4! = 1 ( we know that 0! = 1)

P(X=0) = 1 * (4/5)⁴ = 0.4096

Probability that she will get no questions​ correct = 0.4096

Excel formula :

We use BINOM.DIST(x,n,p,true = Cumulative function or false = Probability Mass function) excel function.

Probability that she will get no questions​ correct = BINOM.DIST(0,4,0.2,FALSE) = 0.4096