A student is taking a true/false exam with 15 questions. If he guesses on each question, what is the probability he will get at least 13 question right?
To find the probability of getting at least 13 questions right on a true/false exam, we need to consider all the possible combinations of answers. Since there are two options (true/false) for each question, there are 2^15 (32,768) possible combinations in total.
Now, let's break down the number of ways the student can get at least 13 questions right.
1. Getting exactly 13 questions right: The student can choose 13 questions out of 15 to answer correctly. This can be computed using the combination formula, which gives us C(15, 13) or 15! / (13! * (15-13)!).
2. Getting exactly 14 questions right: The student can choose 14 questions out of 15 to answer correctly. This can be computed using C(15, 14) or 15! / (14! * (15-14)!).
3. Getting exactly 15 questions right: There is only one way for the student to answer all 15 questions correctly.
To find the probability, we add up the number of ways for each case and divide it by the total number of possible combinations.
Probability = (Number of ways to get 13 questions right + Number of ways to get 14 questions right + Number of ways to get 15 questions right) / Total number of combinations
Probability = (C(15, 13) + C(15, 14) + C(15, 15)) / 2^15
Now, let's calculate the values:
C(15, 13) = 15! / (13! * 2!) = 105
C(15, 14) = 15! / (14! * 1!) = 15
C(15, 15) = 15! / (15! * 0!) = 1
Total number of combinations = 2^15 = 32,768
Probability = (105 + 15 + 1) / 32,768 ≈ 0.00344
Therefore, the probability that the student will get at least 13 questions right is approximately 0.00344 or 0.344%.