Question

A multiple-choice test has 30 questions. There are 4 choices for each question. A student who has not studied for the test decides to answer all the questions randomly by guessing the answer to each question. Which of the following probability distributions can be used to calculate the student’s chance of getting at least 20 questions right ?

• A. Uniform distribution
• B. Exponential distribution
• C. Poisson distribution
• D. Binomial distribution

Hint:

Remember the checklist for a Binomial process (BINS):
Binary outcomes (Success/Failure)
Independent trials
Number of trials is fixed ($n$)
Same probability of success ($p$)

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To model this scenario, we must identify the characteristics of the trials:
1. There is a fixed number of trials ($n = 30$ questions).
2. Each trial has only two mutually exclusive outcomes: success (correct answer) or failure (incorrect answer).
3. The probability of success remains constant across all trials ($p = \frac{1}{4} = 0.25$).
4. The trials are independent (the guess on one question does not affect the guess on another).
Detailed Explanation:
Let us evaluate the suitability of the proposed distributions:
- Uniform distribution: Used when all outcomes are equally likely over a continuous or discrete interval. It does not model a cumulative number of successes.
- Exponential distribution: A continuous distribution used to model time-to-event data.
- Poisson distribution: Used to model the number of events in a continuous time/space interval when the number of trials is extremely large and the success probability is very small.
- Binomial distribution: Perfectly designed for a fixed number of independent Bernoulli trials with a constant probability of success.
Therefore, the Binomial distribution $B(n=30, p=0.25)$ is used to calculate the probability of getting at least 20 questions correct.

Step 2: Final Answer:
The correct choice is the Binomial distribution, matching Option (D).