Rework problem 15 from section 3.5 of your text involving a true-false test. Assume that 20 questions are answered by guessing.
What is the probability of exactly 15 correct answers?
To find the probability of exactly 15 correct answers on a true-false test with guessing, we can use the binomial probability formula. The formula for the probability of getting exactly x successes in n trials is:
P(X = x) = (nCx) * p^x * q^(n-x)
Where:
- P(X = x) represents the probability of getting exactly x successes
- nCx represents the number of ways to choose x items from a set of n items, and it can be calculated using the binomial coefficient formula: nCx = n! / (x! * (n-x)!)
- p represents the probability of success (in this case, the probability of correctly guessing a true-false question)
- q represents the probability of failure (q = 1 - p)
In this problem, you are given that there are 20 questions and they are answered by guessing. In a true-false test, the probability of correctly guessing a question is 1/2, since there are only two possible answers.
Therefore, to find the probability of exactly 15 correct answers, we substitute the given values into the formula:
P(X = 15) = (20C15) * (1/2)^15 * (1/2)^(20-15)
Now, let's calculate the values step by step:
First, calculate the binomial coefficient (20C15):
(20C15) = 20! / (15! * (20-15)!)
This simplifies to:
(20C15) = 20! / (15! * 5!)
Evaluate the factorial values:
20! = 20 * 19 * 18 * 17 * 16 * 15!
15! = 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5!
5! = 5 * 4 * 3 * 2 * 1
Now, substitute these factorial values back into the binomial coefficient formula:
(20C15) = (20 * 19 * 18 * 17 * 16 * 15!) / (15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5!)
= (20 * 19 * 18 * 17 * 16) / (5 * 4 * 3 * 2 * 1)
Simplify the expression:
(20 * 19 * 18 * 17 * 16) = 20,922,240
(5 * 4 * 3 * 2 * 1) = 120
So, (20C15) = 20,922,240 / 120 = 173,355
Next, substitute back into the original formula:
P(X = 15) = (20C15) * (1/2)^15 * (1/2)^(20-15)
= 173,355 * (1/2)^15 * (1/2)^5
= 173,355 * (1/32) * (1/32)
= 173,355 / 32^2
= 173,355 / 1024
≈ 0.169
Therefore, the probability of exactly 15 correct answers on a true-false test with guessing is approximately 0.169, or 16.9%.