Question

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?

Answer 1

We first explain the concept of empirical probability and how the events are considered. We take the given event of getting at least 9 questions correct as event A and U as the universal event. Then we find the number of outcomes. Using the probability theorem of $P\left( A \right)=\dfrac{n\left( A \right)}{n\left( U \right)}$, we get the empirical probability of the events and solve the equation.

Complete step by step answer:
We take two events, one with conditions and other one without conditions. The later one is called the universal event which chooses all possible options. We assume the event of getting at least 9 questions correct as event A and the universal event U as choosing options for the questions and numbers will be denoted as $n\left( A \right)$, $n\left( U \right)$ respectively. Getting 9 correct has two parts - getting exactly 9 questions correct and exactly 10 questions correct.

For exactly 9 correct we have ${}^{10}{{C}_{9}}\times {}^{5}{{C}_{1}}=9\times 5=45$ and for exactly 10 correct we have 1 option. For universal events we have 5 options for every question.So,
$n\left( A \right)=45+1=46$ and $n\left( U \right)={{5}^{10}}$.
We take the empirical probability of the given problem as,
$P\left( A \right)=\dfrac{n\left( A \right)}{n\left( U \right)} \\\ \therefore P\left( A \right) =\dfrac{46}{{{5}^{10}}}$

Therefore,the probability they get at least 9 questions correct is $\dfrac{46}{{{5}^{10}}}$.

Note: We need to understand the concept of universal events. This will be the main event that is implemented before the conditional event. Empirical probabilities, which are estimates, calculated probabilities involving distinct outcomes from a sample space are exact.