Probability help please

Question

Bill has completed a 10-question multiplechoice test on which he answered 7 questions correctly. Each question had
one correct answer to be chosen from five alternatives. Bill says that he answered the test by randomly guessing the answers without reading the questions or answers.


1) Define the random variable x to be the number of correct answers on this test, and construct the probability distribution
if the answers were obtained by random guessing.


2) What is the probability that Bill guessed 7 of the 10 answers correctly?

Answer 1

This is a binomial distribution.  Since every problem has a 20% chance of being guessed right, p = 0.20 and q = 1-0.20=0.80.

 

You would use the formula for binomial distribution to construct the distribution table.  P(r correct) = ₁₀C_r•0.20^r•0.80^(10-r)

For example:

P(0 correct): ₁₀C₀•0.20⁰•0.80^(10-0)=0.107

P(1 correct): ₁₀C₁•0.20¹•0.80^(10-1)=0.268

P(2 correct): ₁₀C₂•0.20²•0.80^(10-2)=0.302

and so on until:

P(10 correct): ₁₀C₁₀•0.20¹⁰•0.80^(10-10)=1.024x10^-7

 

For part 2:

P(7 correct): ₁₀C₇•0.20⁷•0.80^(10-7)=7.864x10^-4