Exploring the Probability of Passing a True/False Exam: A Binomial Distribution Analysis

Imagine a scenario where a student is faced with an important exam containing 17 true or false questions. To pass, the student must answer at least 11 questions correctly. How would one calculate the probability that the student will pass by guessing?

Understanding the Exam Scenarios

This situation can be analyzed using the principles of binomial distribution. In binomial distribution, we are concerned with the probability of obtaining a specific number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure).

Parameters of the Binomial Distribution

The key parameters of this scenario are:

Number of trials (n): This is the total number of questions, which is 17. Probability of success (p): The probability of guessing correctly on each question is 0.5 (since it’s a true or false question). Number of successes (k): We need to find the probability of getting at least 11 correct answers.

Binomial Probability Formula

The probability of getting exactly k successes in n trials is given by the binomial probability formula:

P(X k) binom{n}{k} p^k (1-p)^{n-k}

Where binom{n}{k} is the binomial coefficient, calculated as:

binom{n}{k} frac{n!}{k! (n-k)!}

Calculating the Probability of Passing

To find the probability of passing, i.e., getting at least 11 correct answers, we need to calculate:

P(X ? 11) P(X 11) P(X 12) P(X 13) P(X 14) P(X 15) P(X 16) P(X 17)

Step-by-Step Calculation

For each value of k (11, 12, 13, 14, 15, 16, 17), we can calculate the exact probability:

P(X 11) binom{17}{11} (0.5)^{17} P(X 12) binom{17}{12} (0.5)^{17} P(X 13) binom{17}{13} (0.5)^{17} P(X 14) binom{17}{14} (0.5)^{17} P(X 15) binom{17}{15} (0.5)^{17} P(X 16) binom{17}{16} (0.5)^{17} P(X 17) binom{17}{17} (0.5)^{17}

Calculating the binomial coefficients for each value of k:

binom{17}{11} 12376 binom{17}{12} 6188 binom{17}{13} 2380 binom{17}{14} 618 binom{17}{15} 153 binom{17}{16} 17 binom{17}{17} 1

Summing these coefficients, we get:

12376 6188 2380 618 153 17 1 20713

Now, substituting back into the formula:

P(X ? 11) (12376 6188 2380 618 153 17 1) (0.5)^{17}

P(X ? 11) 20713 (0.5)^{17}

P(X ? 11) ≈ 0.157 or 15.7%

Conclusion

The probability that the student will pass the exam by guessing is approximately 0.157, or 15.7%. This calculation provides a useful insight into the student's chances of success based on pure guesswork.