Model Answer
0 min readIntroduction
Probability theory is a fundamental branch of mathematics dealing with the likelihood of events occurring. In scenarios involving a large number of independent trials, such as a multiple-choice quiz, the binomial distribution provides a powerful tool for calculating probabilities. This question presents a practical application of the binomial distribution, asking us to determine the probability of achieving a specific range of correct answers through random guessing. Understanding this concept is crucial in various fields, including statistics, risk assessment, and decision-making. The problem focuses on 80 questions out of 200, where the student resorts to pure guesswork.
Understanding the Binomial Distribution
The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials. The formula for calculating the probability of exactly k successes in n trials is:
P(X = k) = (n choose k) * pk * (1-p)(n-k)
Where:
- P(X = k) is the probability of exactly k successes
- n is the number of trials
- k is the number of successes
- p is the probability of success on a single trial
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
Applying the Formula to the Question
In this case:
- n = 80 (the number of questions the student guesses on)
- p = 0.25 (the probability of getting a correct answer by guessing, as there are four options)
- We want to find P(25 ≤ X ≤ 30), which means we need to calculate P(X = 25) + P(X = 26) + ... + P(X = 30)
Calculating Individual Probabilities
We need to calculate each P(X = k) for k = 25 to 30 using the formula above. This involves calculating the binomial coefficient and then the probability. Due to the computational complexity, this is best done using statistical software or a calculator with binomial distribution functions. Let's illustrate with P(X=25):
P(X = 25) = (80 choose 25) * (0.25)25 * (0.75)55
Similarly, we calculate P(X = 26), P(X = 27), P(X = 28), P(X = 29), and P(X = 30).
Summing the Probabilities
Once we have calculated each individual probability, we sum them up to get the final probability:
P(25 ≤ X ≤ 30) = P(X = 25) + P(X = 26) + P(X = 27) + P(X = 28) + P(X = 29) + P(X = 30)
Approximation using Normal Distribution
Since n is large (n=80), we can approximate the binomial distribution with a normal distribution. The mean (μ) and standard deviation (σ) of the binomial distribution are:
- μ = n * p = 80 * 0.25 = 20
- σ = √(n * p * (1-p)) = √(80 * 0.25 * 0.75) = √15 ≈ 3.87
We can then use the normal distribution to approximate the probability. We need to apply a continuity correction. We want P(25 ≤ X ≤ 30), which becomes P(24.5 < X < 30.5) in the normal approximation.
We calculate the Z-scores for 24.5 and 30.5:
- Z1 = (24.5 - 20) / 3.87 ≈ 1.19
- Z2 = (30.5 - 20) / 3.87 ≈ 2.72
Then, P(24.5 < X < 30.5) = P(Z < 2.72) - P(Z < 1.19). Using a standard normal distribution table, P(Z < 2.72) ≈ 0.9967 and P(Z < 1.19) ≈ 0.8830. Therefore, P(24.5 < X < 30.5) ≈ 0.9967 - 0.8830 ≈ 0.1137.
Therefore, the probability that sheer guesswork yields from 25 to 30 correct answers for 80 of the 200 questions is approximately 0.1137 or 11.37%.
Conclusion
In conclusion, using the binomial distribution (or its normal approximation), we've determined the probability of achieving between 25 and 30 correct answers through random guessing on 80 questions. The approximate probability of 11.37% suggests that while not highly likely, it's certainly a plausible outcome given the number of trials. This illustrates the power of probability distributions in analyzing random events and quantifying uncertainty. Further refinement of the calculation would require computational tools for precise binomial coefficient calculations.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.