Question

Emmanuel is given a multiple-choice exam with ten questions and each question with five possible answers. He decided to guess randomly for each question. What is the
probability that he will get at least six questions correct?

Answer 1

Let $n=10$ (number of questions) and $p=1/5$ (the probability to choose correct answer).

Now we can consider $X$ - number if correct answers, and $X\sim Bin(10,1/5)$

We need to find $P(X\geq 6)$ - the probability that he will get at least six questions correct.

For $X\sim Bin(n,p)$ we have: $P(X=k)=\binom n k p^k (1-p)^{n-k}$

$P(X\geq 6)=P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)=210 \times(1/5)^6 (4/5)^4+120\times (1/5)^7(4/5)^3+45\times (1/5)^8(4/5)^2+10\times (1/5)^94/5+1\times(1/5)^{10}\approx 0.0064$

Answer: 0.0064