Final answer:
The probability of passing the true-false quiz with at least a 70 percent score when guessing randomly can be found using the binomial probability formula to sum the probabilities of getting exactly 7, 8, 9, or 10 questions right.
Step-by-step explanation:
The student seeks to understand the probability of passing a quiz by correctly answering at least 70% of the questions when guessing randomly on a true-false test. To pass the test, the student must answer correctly 7 out of 10 questions, as 70% of 10 is 7. For each question, there are two possible outcomes: true or false, so the probability of guessing correctly is 1/2 for each question.
To calculate the probability of getting at least 7 answers correct, we must consider the probabilities of getting exactly 7, 8, 9, or all 10 questions correct. This can be calculated using the binomial probability formula:
P(X = k) = (n choose k) * (p)^k * (1-p)^(n-k)
where 'P(X = k)' is the probability of getting exactly 'k' questions correct, 'n' is the total number of questions (10 in this case), 'p' is the probability of guessing a single question correctly (1/2), and '(n choose k)' is the binomial coefficient.
The final probability is the sum of the probabilities for 7, 8, 9, and 10 correct answers, which would be:
P(X ≥ 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
By calculating each component using the binomial probability formula, we can find this cumulative probability.