Question

A student taking a 10 -question, true-false diagnostic test knows none of the answers and must guess at each one. Compute the probability that the student gets a score of 80 or higher. What is the probability that the grade is 70 or lower?

Short answer

The probability of scoring 80 or higher is 0.055; scoring 70 or lower is 0.945.

Step-by-step solution

Define the Problem

We need to find the probability of a student getting a score of 80 or higher and 70 or lower on a 10-question true-false test by guessing. Each question has two options: true or false. Thus, the probability of guessing a question correctly is 0.5.

Model the Problem with a Binomial Distribution

The problem can be modeled as a binomial distribution, where the number of trials (n) is 10 (the number of questions), and the probability of success (getting a question right) is 0.5. The binomial distribution describes the number of successes in n independent trials.

Calculate the Probability of Scoring 80 or Higher

A score of 80 on 10 questions implies getting at least 8 out of 10 questions correct. We need to find the probability P(X ≥ 8), where X is the number of correct answers from n = 10 questions. This equals the sum of probabilities P(X = 8), P(X = 9), and P(X = 10). Use the binomial probability formula: \[ P(X = k) = \binom{n}{k} (p)^k (1-p)^{n-k} \]where \( \binom{10}{k} \) is the binomial coefficient, and \( p = 0.5 \). Calculate these for k = 8, 9, and 10.

Calculations for k = 8, 9, 10

Calculate each probability: - P(X = 8): \[ \binom{10}{8} (0.5)^8 (0.5)^2 = 45 \times (0.5)^{10} \approx 0.044 \]- P(X = 9): \[ \binom{10}{9} (0.5)^9 (0.5)^1 = 10 \times (0.5)^{10} \approx 0.01 \]- P(X = 10): \[ \binom{10}{10} (0.5)^{10} = 1 \times (0.5)^{10} \approx 0.001 \]

Total Probability for Scoring 80 or Higher

Sum the probabilities calculated for X = 8, 9, and 10:\[ P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) \approx 0.044 + 0.01 + 0.001 = 0.055 \]

Calculate the Probability of Scoring 70 or Lower

A score of 70 implies getting 7 or fewer questions correct, so we find the probability P(X ≤ 7). This is equivalent to 1 - P(X ≥ 8). From the previous step, we found that P(X ≥ 8) = 0.055. Therefore:\[ P(X \leq 7) = 1 - P(X \geq 8) = 1 - 0.055 = 0.945 \]

Key concepts

Understanding Probability Theory

Probability theory is the mathematical framework for quantifying the likelihood of different outcomes. In this context, it serves as the foundation for determining how likely it is for a student to score a certain percentage on a test by random guessing.

Key concepts in probability theory include:

• Randomness: Each question guessed by the student is an independent trial, meaning the outcome of one trial does not affect the others.
• Probability: The likelihood of guessing a question correctly is fixed at 0.5 (since there are equal chances of choosing either true or false).
• Events: We want to calculate the probability of events such as scoring 80% (8 correct answers) or more and 70% (7 correct answers) or less.

Probability theory simplifies complex scenarios by turning them into quantifiable predictions, allowing us to use tools like the binomial distribution to make calculations.

Combinatorics in Binomial Distribution

Combinatorics is a mathematical technique that deals with counting and arranging possibilities. It plays a crucial role in calculating probabilities in scenarios like our true-false test.

For example, when determining the probability of getting exactly 8 out of 10 questions correct, we need to consider all possible ways 8 questions can be selected as correct. This is where the binomial coefficient comes in, a key combinatorial concept represented as \( \binom{n}{k} \), which calculates the number of ways to choose \( k \) successes out of \( n \) trials.

The binomial distribution relies on:

• Binomial Coefficient: \( \binom{10}{8} \) for our test means there are 45 ways to pick 8 questions that can be correct.
• Ordered Outcomes: A binomial outcome relies on both the chosen combinations and the probabilities of those combinations occurring.

Combinatorics provides the framework to pair probabilities with possible combinations, helping compute the results we need.

Statistical Modeling Through Binomial Distribution

Statistical modeling involves using mathematical models to represent data or scenarios, often to predict an outcome or understand an underlying process. In our exercise, the scenario is modeled using a binomial distribution.

The binomial distribution is ideal for situations with two possible outcomes (success or failure). Here, each question is a trial with a binary outcome (correct or incorrect).
The parameters of the binomial model are:

• Number of trials \( n \): 10, as there are 10 questions on the test.
• Probability of success \( p \): 0.5, the chance of guessing a question correctly.

The binomial formula \( P(X = k) = \binom{n}{k} (p)^k (1-p)^{n-k} \) calculates the probability \( P \) of getting exactly \( k \) successes (correct answers) among \( n \) trials.

Thus, statistical modeling helps transform a complex guessing scenario into a structured probabilistic problem, providing insights into likely outcomes and their probabilities.