Problem 45 - Calculus for Biology and Medicine

Question

A multiple-choice exam contains 50 questions. Each question has four choices. Find the expected number of correct answers if a student guesses the answers at random.

Step-by-Step Solution

Answer

The expected number of correct answers is 12.5.

Step 1: Determine Probability of Correct Answer for One Question

Since each question has four choices and only one is correct, the probability of guessing the correct answer for any single question is \( \frac{1}{4} \).

Step 2: Identify Number of Trials and Expected Value Formula

There are 50 questions on the exam, representing 50 trials. For a binomial distribution, the expected number of successes (correct answers) can be calculated using the formula \( E(X) = n \cdot p \), where \( n \) is the number of trials and \( p \) is the probability of success on each trial.

Step 3: Calculate Expected Number of Correct Answers

Substitute the values \( n = 50 \) and \( p = \frac{1}{4} \) into the expected value formula to find the expected number of correct answers. Thus, \[ E(X) = 50 \cdot \frac{1}{4} = 12.5. \]

Key Concepts

Expected Value

The concept of the expected value is a fundamental principle in statistics and probability. It essentially provides a measure of the center of a probability distribution, often referred to as the "long-run average". In other words, it is what you would expect to happen on average if you were to repeat a random experiment multiple times.

When evaluating the expected value in a situation involving multiple trials, such as in our multiple-choice example, it helps to predict what the average outcome will be. This is calculated using the formula:

• \( E(X) = n \cdot p \)

where:

• \( n \) is the number of trials.
• \( p \) represents the probability of success on a single trial.

In our exercise, there are 50 questions, each with four possible answers, and each guess has a probability of \( \frac{1}{4} \).
This allows us to compute the expected number of correct answers as \( 50 \cdot \frac{1}{4} = 12.5 \).
Hence, a student guessing randomly expects to get around 12.5 questions correct on average.

Probability

Probability is the likelihood or chance of an event occurring and is a crucial concept in understanding randomness and uncertainty in various experiments or real-life scenarios. It is meant to quantify the degree of certainty of an event, expressed as a fraction from 0 to 1, where 0 indicates an impossible event, and 1 indicates a certain event.

In any given trial, the probability of a specific outcome can be calculated. For our multiple-choice problem, each question has four options, only one of which is correct. Therefore, the probability of selecting the correct answer by pure guessing is:\( \frac{1}{4} \).

Each question acts as an independent trial, meaning the probability of guessing each question right does not change based on previous guesses. This independence is crucial because it allows us to apply the rules of the binomial distribution, making the calculations for expected value straightforward and systematic.
Understanding probability helps frame expectations and observe how likely certain outcomes are in comparison to others.

Random Guessing

Random guessing is when an individual makes a choice without any prior knowledge or strategy, relying purely on chance. In our multiple-choice example, every guess is independent, akin to tossing a fair four-sided die for each question.

This essentially means that instead of using knowledge or logic to answer questions, the student selects answers at random. The probability of getting a question correct remains constant at \( \frac{1}{4} \) due to the random nature and the uniform chance across all options.
When analyzing random guessing, it's fascinating to observe how statistical tools like the expected value give us insight into the likely outcome over a large number of trials.
Through random guessing, although any one guess is unlikely to be correct, over a series of many questions, the law of large numbers suggests the average results will converge to their expected value. Hence, with 50 questions, it's anticipated that the student will get around 12 or 13 correct, probabilistically aligning with the prediction of 12.5.