Probability problem

[raffaele2406]

Hello everybody,
I tried to solve this problem:

In a close answers quiz there are 8 items each of which can have 4 possible answers A, B, C and D and only one answer is correct.
A quiz result is considered positive if at least 4 items have correct answers. What is the probability to get a positive result if someone answers randomly?

I made a small program which simulates the situation and I got about 0.113
Too bad I can't solve it using the formulas.

Any help will be appreciated
Thank you in advance

 

[Romsek]

The probability of correctly answering a question when randomly answering is \(\displaystyle p = \dfrac 1 4\)
There are 8 questions so the number of questions correctly answered out of 8 has a binomial distribution with parameters
\(\displaystyle p=\dfrac 1 4,~n=8\)

\(\displaystyle P(k) = \dbinom{8}{k}\left(\dfrac 1 4\right)^k \left(\dfrac 3 4\right)^{8-k}\)

\(\displaystyle P(\text{positive result)} = P(N\geq 4) = 1 - P(N < 4) = 1-P(0) - P(1) - P(3)\)

I leave you to the plug and chug.