# Probability problem

#### raffaele2406

##### New member
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

#### Romsek

##### Full Member
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.