# A 10 question multiple choice exam is given and each question has 5 possible answers. A student takes this exam and guesses at every question. What is the probability they get at least 9 questions correct?

$0.0000041984 \approx 4.2 \times {10}^{- 6}$

#### Explanation:

Let's first set up the binomial. The general formula is:

sum_(k=0)^(n)C_(n,k)(p)^k((~p)^(n-k))

We have $n = 10$.

With 5 possible answers on each question, this gives the probability of guessing the correct answer $p = \frac{1}{5}$, meaning the probability of getting it wrong is ~p=4/5.

We're only looking at the probability of getting at least 9 questions correct, and so only care about getting 9 questions correct and 10 questions correct. This gives:

${C}_{10 , 9} {\left(\frac{1}{5}\right)}^{9} {\left(\frac{4}{5}\right)}^{1} + {C}_{10 , 10} {\left(\frac{1}{5}\right)}^{10} {\left(\frac{4}{5}\right)}^{0}$

$\left(10\right) {\left(\frac{1}{5}\right)}^{9} {\left(\frac{4}{5}\right)}^{1} + \left(1\right) {\left(\frac{1}{5}\right)}^{10} {\left(\frac{4}{5}\right)}^{0} = 0.0000041984 \approx 4.2 \times {10}^{- 6}$ or roughly 3X more likely than being hit by lightening over the course of a year.