Problem 27 - Algebra 2

Question

Statistics A multiple-choice test has ten questions. Each question has five choices, with only one correct. Statisticians consider a "rare" event to have less than a 5\(\%\) chance of occurring. According to this standard, what grades would be rare on this test if you guess? Justify your answer.

Step-by-Step Solution

Answer

A rare grade is defined as having less than a 5\(\%\) chance of occurring. After calculating the binomial probabilities for grades 0 through 10, it is found that grades 0 (0 correct answers), 1, 9, and 10 (10 correct answers) have less than 5\(\%\) probability of occurring when guessing all answers, and thus are considered 'rare' grades.

Step 1: Understand the Probability of Each Outcome

Since each question is a multiple-choice question with five options, the chance of guessing the correct answer to any one question is \(\frac{1}{5}\) or 20\(\%\). The performance on each question can be modeled as a Bernoulli trial, which is a random experiment with exactly two possible outcomes ('success'= guessing the correct answer and 'failure'= guessing the wrong answer). This makes the entire test a binomial experiment, i.e. a series of n identical Bernoulli trials.

Step 2: Check Different Grades for 'Rarity'

Now start calculating the binomial probability of getting a certain number of correct answers (ranging from 0 to 10). The formula used here is the binomial probability formula which is: \[ P(X=k) = C(n, k) * (p)^k * (1-p)^{n-k} \] where \( P(X=k) \) is the probability of getting k successes in n trials, \( C(n, k) \) is the binomial coefficient or number of combinations of n items taken k at a time, p is the probability of success on a single trial, and \( (1-p) \) is the probability of failure on a single trial.

Step 3: Define Rare Grades

After calculating the binomial probabilities for various grades, next find out which probability is less than 5\(\%\). Any grade (number of correct answers) whose probability of occurrence is less than 5\(\%\) is considered a 'rare' grade.

Key Concepts

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the chances of achieving a particular number of successes in a specific number of identical and independent trials, where each trial has two possible outcomes. In our multiple-choice test example, each question can either be a success (correct answer) or a failure (incorrect answer). This setup fits perfectly with the concept of a binomial distribution.

• Each trial (question) has the same probability of success.
• Trials are independent; the outcome of one does not affect the others.
• The number of trials (questions) is fixed.
• The binomial random variable, often denoted as \( X \), represents the number of successes in \( n \) trials.

In our example, the binomial distribution helps evaluate the likelihood of guessing a certain number of questions correctly by chance alone.

Bernoulli Trials

A Bernoulli trial is a random experiment with exactly two possible outcomes; traditionally called "success" and "failure." In our test scenario, answering a question correctly is considered a success, while an incorrect answer is deemed a failure. Some characteristics of Bernoulli trials include:

• Each trial is independent.
• There are only two outcomes: success (correct) and failure (incorrect).
• Probability of success is constant across trials.

Since each question in the test can be represented by a Bernoulli trial, the entire test, consisting of several Bernoulli trials, combines into what is known as a binomial experiment. This forms the basis for applying the binomial distribution to calculate probabilities.

Binomial Probability Formula

The binomial probability formula calculates the probability of achieving exactly \( k \) successes in \( n \) independent Bernoulli trials. You can think of it as the tool that helps us predict outcomes for scenarios like guessing answers on a test.The formula is expressed as:\[ P(X=k) = C(n, k) \times p^k \times (1-p)^{n-k} \]Here's what each part means:

• \( P(X=k) \): Probability of getting exactly \( k \) successes (correct answers).
• \( C(n, k) \): Binomial coefficient, the number of combinations for choosing \( k \) successes out of \( n \) trials, which can be calculated as \( \frac{n!}{k!(n-k)!} \).
• \( p \): Probability of success on a single trial, here \( \frac{1}{5} \) or 20% for each question.
• \( (1-p) \): Probability of failure, or 80% in this context.
• \( n \): Total number of trials, which is 10 for our test.
• \( k \): Number of successes.

This formula is essential for determining which scores are rare, hence less than a 5% chance, when randomly guessing.