There is a multiple choice exam with 24 questions, and five options each. If i guess every question, what is the probability that i will get exactly 15 correct.
To calculate the probability of getting exactly 15 correct answers out of 24 questions with five options each, we can use the binomial probability formula.
The formula is given by:
P(X=k) = (nCk) * (p^k) * (q^(n-k))
Where:
- P(X=k) is the probability of getting exactly k successes (in this case, 15 correct answers)
- n is the total number of trials (in this case, the total number of questions, which is 24)
- k is the number of successful outcomes or correct answers (in this case, 15)
- nCk represents the binomial coefficient, also known as "n choose k." It calculates the number of possible combinations of k items from a set of n items, which can be calculated as n! / (k! * (n-k)!)
- p is the probability of success on an individual trial (in this case, the probability of guessing the correct answer, which is 1/5)
- q is the probability of failure on an individual trial (in this case, the probability of guessing incorrectly, which is 1 - p)
Now, let's substitute the values into the formula and calculate:
P(X=15) = (24C15) * ((1/5)^15) * ((1 - 1/5)^(24-15))
To calculate (24C15), we'll use the binomial coefficient formula:
(24C15) = (24!) / (15! * (24-15)!)
Calculating (24C15) gives us:
(24C15) = (24!) / (15! * 9!)
Using factorials, this simplifies to:
(24C15) = (24 * 23 * 22 * 21 * 20 * 19 * 18 * 17 * 16) / (15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
Now, we can substitute this value and the other values into the binomial probability formula and calculate the probability of getting exactly 15 correct answers.