Question

I do not understand the binomial theorem. One of my questions is "there are 5 mutiple choice questions with 4 possible answers each. What is the probability of getting more than 3, exactly 3, and less than 3 correct?"
Thanks for any help:)

Answer 1

Use the binomial theorem to get the possibility of 0,1,2,3,4 and 5 correct.
For zero correct, the probability is just (3/4)^5 = 0.2373. 3/4 is the probability of getting each one wrong.
For five correct, the probability is
(1/4)^5 = 0.0001
For one correct, (one success and 4 failures) the probability is
5!/[1!*4!]*(1/4)*(3/4)^4
= 5(.25)(.3164) = 0.3955
(That is where you need the binomial theorem)
For two correct, using the same theorem, the probability is
[5!/(3!*2!)](1/4)^2*(3/4)^3
= 10*(0.25)^2(0.4219)= 0.2637
For three correct, the probability is
5!/[2!*3!)](1/4)^3*(3/4)^2 = 0.0879
For four correct, the probability is
(5!/4!)(1/4)^4*(3/4)= 5*.0039*.3164 = 0.0195

Use these results to get the probabilities for >3 and <3 right.

Answer 2

Use binomial the porem to expand (x+y) 25