Exercises: Develop a Theoretical Probability Distribution and Find Its Expected Value

On a multiple-choice test each question has four choices and a guesser is correct with
probability $\frac{1}{4}$. The questions are independent. What is the probability of getting
the first two specific questions both correct?

A student guesses on every question of a five-question test where each question has four
equally likely choices. Let $X$ be the number of questions answered correctly. Which statement
correctly describes the model?

How many different ways are there to choose which 3 of the 5 questions are the correct ones?
That is, evaluate $\binom{5}{3}$.

For the five-question guessing test ($P(\text{correct}) = \frac{1}{4}$), which expression
correctly gives $P(X = 2)$, the probability of exactly 2 correct?

For the five-question guessing test, compute $P(X = 2) = \binom{5}{2}\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^3$.
Write your answer as a fraction over a denominator of $1024$.

For the five-question guessing test, compute $P(X = 3) = \binom{5}{3}\left(\frac{1}{4}\right)^3\left(\frac{3}{4}\right)^2$.
Write your answer as a fraction over a denominator of $1024$.

The full developed distribution for the five-question guessing test is shown below.

Add the six numerators to verify the distribution is valid. What is the sum of the numerators?

Using the distribution table above, compute the expected number correct
$E(X) = \sum k \cdot P(X=k)$ for the five-question guessing test. Give a decimal.