Probability of Passing a True/False Quiz with Random Guessing

In this article, we will explore the probability that a student who does not study for a ten-question true/false quiz and guesses randomly will pass the quiz with a grade of at least 70% correct answers. This involves understanding the binomial distribution and calculating the relevant probabilities.

Step 1: Determine the Passing Score

To pass the quiz with a grade of at least 70%, the student needs to answer at least 7 out of 10 questions correctly. Therefore, the passing number of correct answers is 7, 8, 9, or 10.

Step 2: Model the Problem

The problem can be modeled using a binomial distribution. The parameters for the binomial distribution are:

n 10, the total number of questions p 0.5, the probability of guessing a question correctly

The number of correct answers, X, follows a binomial distribution:

X ~ Binomial(n10, p0.5)

Step 3: Calculate the Probability of Passing

We need to find the probability of obtaining at least 7 correct answers out of 10, which is the sum of the probabilities of getting exactly 7, 8, 9, or 10 correct answers.

The probability of getting exactly k correct answers in a binomial distribution is given by the formula:

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

Step 4: Calculate Each Probability

We will calculate the probability for each case:

For k 7: For k 8: For k 9: For k 10:

For k 7:

P(X 7) binom{10}{7} (0.5)^7 (1-0.5)^{10-7} binom{10}{7} (0.5)^{10}

binom{10}{7} 120

P(X 7) 120 times frac{1}{1024} frac{120}{1024} frac{15}{128}

For k 8:

P(X 8) binom{10}{8} (0.5)^8 (1-0.5)^{10-8} binom{10}{8} (0.5)^{10}

binom{10}{8} 45

P(X 8) 45 times frac{1}{1024} frac{45}{1024}

For k 9:

P(X 9) binom{10}{9} (0.5)^9 (1-0.5)^{10-9} binom{10}{9} (0.5)^{10}

binom{10}{9} 10

P(X 9) 10 times frac{1}{1024} frac{10}{1024} frac{5}{512}

For k 10:

P(X 10) binom{10}{10} (0.5)^{10} (1-0.5)^{10-10} binom{10}{10} (0.5)^{10}

binom{10}{10} 1

P(X 10) 1 times frac{1}{1024} frac{1}{1024}

Step 5: Sum the Probabilities

The total probability of passing is the sum of the probabilities for k 7, 8, 9, and 10:

P(X geq 7) P(X 7) P(X 8) P(X 9) P(X 10)

P(X geq 7) frac{120}{1024} frac{45}{1024} frac{10}{1024} frac{1}{1024}

P(X geq 7) frac{176}{1024} frac{11}{64}

Conclusion

The probability that the student passes the quiz by answering at least 70% of the questions correctly is:

boxed{frac{11}{64}} approx 0.171875 text{ or } 17.19%.