11. A Certification Exam containing 20 true and false questions. If a student guesses at the answer to each question, what is the probability that she/he will get exactly 14 correct?
pr=(1/2)^20
To calculate the probability of getting exactly 14 correct answers out of 20 true or false questions, we need to use the binomial probability formula.
The binomial probability formula is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) represents the probability of getting exactly k successes
- C(n, k) is the binomial coefficient, calculated as C(n, k) = n! / (k! * (n-k)!)
- n is the total number of trials or questions (20 in this case)
- k is the number of desired successful outcomes (14 in this case)
- p is the probability of success on a single trial, which is 1/2 for a true or false question
Now, let's calculate the probability:
P(X = 14) = C(20, 14) * (1/2)^14 * (1 - 1/2)^(20-14)
Calculating the binomial coefficient:
C(20, 14) = 20! / (14! * (20-14)!)
= 20! / (14! * 6!)
Let's simplify this expression:
P(X = 14) = (20! / (14! * 6!)) * (1/2)^14 * (1/2)^6
Now, calculate the probability:
P(X = 14) = (20*19*18*17*16*15*14! / (14! * 6!)) * (1/2)^20
Simplifying further:
P(X = 14) = (20*19*18*17*16*15) / (6! * 2^20)
Calculating this expression:
P(X = 14) ≈ 0.1841
Therefore, the probability of getting exactly 14 correct answers by guessing on a 20-question true or false exam is approximately 0.1841 or 18.41%.