Random Guessing on Multiple Choice Exams: A Statistical Analysis

In an upcoming examination, there will be 20 multiple choice questions with five choices for each question. If a student was to randomly select answers, how many questions would be answered correctly? This article delves into examining the expected number of correct answers and simulates the process of random guessing several times. We will also look at the probability of answering a specific number of questions correctly.

Expected Number of Correct Answers

For each question, the probability of selecting the correct answer is 1/5, since there are 5 choices. Hence, the expected number of correct answers for 20 questions can be calculated as:

Expected Correct Answers Number of Questions times; Probability of Correct Answer 20 times; 1/5 4

Simulation

To understand the variability of correct answers when completely guessing, we can simulate the process of selecting answers multiple times. Here’s a simple simulation in Python:

import randomdef simulate_exam(num_questions20, num_choices5, num_simulations5):    results  []    for _ in range(num_simulations):        correct_answers  0        for _ in range(num_questions):            # Randomly select an answer (0 for incorrect, 1 for correct)            if random.randint(1, num_choices)  1:                correct_answers   1        (correct_answers)    return results

Running the Simulation:

simulation_results  simulate_exam()simulation_results

Example Simulation Results:

If we were to run the simulation, we might get results like:

Simulation 1: 3 correct answers Simulation 2: 5 correct answers Simulation 3: 4 correct answers Simulation 4: 2 correct answers Simulation 5: 4 correct answers

Summary: Based on the expected value, a student randomly guessing would average around 4 correct answers, but actual results can vary due to the randomness of the selections. Our example provided simulated results showing a range of correct answers from 2 to 5.

Calculating the Probability of Correct Answers

Alternatively, we can use the binomial distribution to calculate the probability of answering a specific number of questions correctly. Let X be the number of correct guesses:

X ~ B(20, 0.2)

The probability of answering exactly 8 questions correctly is:

P(X8) C(20, 8) times; 0.2^8 times; 0.8^17 ≈ 0.06235

Here, 1/5 chance of guessing 1 question right and there are 8 questions, so the chance is 1/5^8 which would be 1/390625 ≈ 2.56 x 10^-6. Converting that to a percentage results in 2.56 x 10^-4 or 0.000256.

Therefore, the probability of getting a specific number of questions correct by random guessing can be calculated using the binomial formula.

Conclusion: Random guessing on multiple choice exams is a topic rich in probability and expected value calculations. These principles offer insights into the variability and predictability of outcomes, which can help students and educators alike understand the nature of chance and probability.