What is the probability of getting at least 50 questions of 100 right?

Question

If there are 100 MCQs with 4 options each. The probability that a person gets an question right is 0.25. What is the probability of getting at least 50 questions of 100 right?

Answer 1

Hint: The number of ways to get $n$ questions right is $\binom{100}{n}$, and given any set of $n$ questions to get right the probability of doing so is

$$(0.25)^n(0.75)^{100-n}.$$

Can you solve it from there?

Answer 2

The number $X$ of questions correct if the student is answering strictly at random (with no knowledge of the subject matter) has $X \sim \mathsf{Binom}(n = 100,\, p = 1/4).$

So the probability of getting exactly fifty questions correct is $$P(X = 50) = {100 \choose 50}\left(\frac 14\right)^{50}\left(\frac34\right)^{50} = 4.5073 \times 10^{-8}.$$ Computation using R statistical software:

dbinom(50, 100, 1/4)
## 4.507311e-08

The probability of getting at least fifty correct is

$$P(X \ge 50) = \sum_{k=50}^{100} {100 \choose k}\left(\frac 14\right)^{k}\left(\frac34\right)^{100-k} = 6.6385 \times 10^{-8}.$$

1 - pbinom(49, 100, 1/4)
## 6.638502e-08

Both probabilities are very small because most of the probability in the distribution $\mathsf{Binom}(n = 100,\, p = 1/4)$ is centered near $\mu = E(X) = np = 100(1/4) = 25.$

Here is a figure that shows the distribution of $\mathsf{Binom}(n = 100,\, p = 1/4)$ along with the density function of $\mathsf{Norm}(\mu = 25, \sigma = 4.33),$ where $\sigma = \sqrt{np(1-p)} =$ $\sqrt{75/4} = 4.3301.$

Unless you are using software (or a statistical calculator) in your class, my guess is that you are supposed to use the normal approximation to the binomial distribution to approximate the very small value of $P(X \ge 50).$ (As @LordShark commented.) I will show the start of that procedure, and let you verify it and finish it for yourself:

$$P(X \ge 50) = P(X > 49.5) = P\left(\frac{X - \mu}{\sigma} > \frac{49.5 - 25}{4.3301} \right)\\ \approx P(Z > 5.658) = ?$$ where $Z$ is a standard normal random variable.