A student guesses blindly on a 20-question multiple choice test. If each question has 4 answer choices (only one of which is correct), what is the probability that the student will answer exactly 10 questions correctly?
To find the probability that the student will answer exactly 10 questions correctly, we need to use the concept of binomial probability.
The binomial probability formula is:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X=k) is the probability of getting exactly k successes
C(n, k) is the number of combinations of choosing k successes from n trials
p is the probability of success on a single trial
n is the total number of trials
In this case, k = 10 (the student wants to answer exactly 10 questions correctly), p = 1/4 (since there is only 1 correct answer out of 4 choices), and n = 20 (the total number of questions).
Now, let's plug the values into the formula:
P(X=10) = C(20, 10) * (1/4)^10 * (3/4)^(20-10)
To calculate the combination (C) function, we use the formula:
C(n, k) = n! / (k! * (n-k)!)
Let's calculate C(20, 10):
C(20, 10) = 20! / (10! * (20-10)!)
= 20! / (10! * 10!)
We can simplify this by canceling out some terms:
C(20, 10) = (20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11) / (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
Now, let's calculate the probability:
P(X=10) = (20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11) / (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) * (1/4)^10 * (3/4)^10
Using a calculator, we can compute this:
P(X=10) ≈ 0.1762
Therefore, the probability that the student will answer exactly 10 questions correctly is approximately 0.1762, or about 17.62%.