A student takes a true-false test that has 8 questions and guesses randomly at each answer. Let X be the number of questions answered correctly. Find P(5)
Disregarding the number of answer choices for each question, then P(5)=5/8=0.625=62.5%
To find the probability of getting 5 questions correct, we need to calculate the number of ways that the student can get exactly 5 out of 8 questions correct and then divide it by the total number of possible outcomes.
Since each question has only two possible answers (true or false), there are 2^8 = 256 possible outcomes in total. This is because for each question, the student can either answer correctly (1 possibility) or answer incorrectly (1 possibility), and there are 8 questions in total. Thus, the total number of possible outcomes is 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 256.
To calculate the number of ways to get exactly 5 out of 8 questions correct, we can use the binomial coefficient formula C(n, k), which calculates the number of combinations of choosing k items from a set of n items. In this case, n represents the total number of questions (8), and k represents the number of questions answered correctly (5).
Using the binomial coefficient formula, we can calculate C(8, 5) as follows:
C(8, 5) = 8! / (5!(8-5)!) = (8 * 7 * 6) / (3 * 2 * 1) = 56
Therefore, there are 56 ways in which the student can get exactly 5 out of 8 questions correct.
To find the probability P(5), we divide the number of ways to get 5 questions correct (56) by the total number of possible outcomes (256):
P(5) = 56 / 256 = 0.21875
Therefore, the probability of the student answering exactly 5 out of 8 questions correctly is approximately 0.21875, or 21.875%.