Final answer:
To set a passing grade for a 15-question true-false quiz to ensure that the probability of a student guessing and passing is less than 0.1, one must calculate the cumulative binomial probability of scoring various grades and set the passing grade accordingly. The lowest passing grade is achieved when the cumulative probability of scoring that grade or lower is less than 0.1. The specific passing grade and the probability of passing by guessing would require calculations using binomial probabilities or statistical software.
Step-by-step explanation:
To determine the lowest passing grade on a 15-question true-false quiz so that the probability of a student guessing and passing is less than 0.1, we must first understand the binomial probability distribution. Here, each question has two possible outcomes, true or false, with a probability of 0.5 for a student guessing the answer correctly. To find the lowest passing grade, we look for the smallest number of correct answers for which the cumulative probability of getting that many or fewer correct answers by guessing is less than 0.1.
Using binomial probability formulas or a calculator with statistical functions, we calculate the probabilities for each possible number of correctly guessed answers. To pass with at least 70 percent, a student would need to get at least 11 out of 15 questions correct (since 15 × 0.7 = 10.5, and we round up). We sum the probabilities of getting 0 to 10 correct answers to ensure that this sum is less than 0.1. If the sum is less than 0.1, then 11 is the lowest passing grade.
Once we have calculated that 11 is the lowest passing score that meets the professor's criteria, we proceed to calculate the probability of a student guessing and obtaining a score of exactly 11, 12, 13, 14, or 15 correct answers and add those probabilities together to find the total probability of passing by guessing.