A student takes a true-false test that has 13 questions and guesses randomly at each answer. Let X be the number of questions answered correctly. Find P(8)
Well, since the student is guessing randomly at each answer, we can assume that the probability of getting each question right is 0.5 (for true-false questions). So, we can determine the probability of the student getting exactly 8 questions correct using the binomial probability formula:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
In this case, n = 13 (the number of questions) and p = 0.5 (the probability of getting a question right).
Plugging in the values:
P(8) = (13C8) * (0.5)^8 * (1-0.5)^(13-8)
Calculating:
P(8) = (13!/((13-8)! * 8!)) * (0.5)^8 * (0.5)^5
P(8) = (13!/5!8!) * (0.5)^13
P(8) = (1287) * (0.00012207031)
P(8) ≈ 0.157
So, the probability of the student getting exactly 8 questions correct is approximately 0.157.
To find the probability of getting exactly 8 questions correct, we use the binomial probability formula:
P(X = k) = nCk * p^k * (1-p)^(n-k)
Where:
- n is the total number of trials (number of questions in this case)
- k is the number of successful events (number of correct answers in this case)
- p is the probability of success (the chance of guessing a correct answer)
In this case, the student has a 50% chance of guessing a correct answer for each question since it is a true-false test (p = 0.5), and there are 13 questions (n = 13). We want to find P(X = 8), so k = 8.
Plugging these values into the formula:
P(8) = 13C8 * 0.5^8 * (1-0.5)^(13-8)
Using the binomial coefficient formula:
nCk = n! / (k! * (n-k)!)
Calculating nCk:
13C8 = 13! / (8! * (13-8)!)
= 13! / (8! * 5!)
= (13 * 12 * 11 * 10 * 9 * 8!) / (8! * 5!)
= 13 * 12 * 11 * 10 * 9 / (5 * 4 * 3 * 2 * 1)
= 13 * 12 * 11 * 10 * 9 / 120
= 154,440 / 120
= 1,287
Now plugging all the values into the binomial probability formula:
P(8) = 1,287 * (0.5^8) * (0.5^5)
= 1,287 * 0.00390625 * 0.03125
= 1,287 * 0.0001220703125
= 0.157
Therefore, P(8) is approximately 0.157, or 15.7%.
To find P(8), we need to determine the probability of getting exactly 8 questions correct out of the 13 questions on the test when guessing randomly.
Since each question has two possible outcomes (true or false), and the student is guessing randomly, the probability of guessing the correct answer to any particular question is 1/2.
Now, we can use the binomial probability formula to calculate the probability of getting exactly 8 questions correct out of 13:
P(X=k) = C(n,k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of getting k successes (in our case, correct answers),
- n is the total number of trials (13 questions),
- k is the number of successes (8 correct answers),
- p is the probability of success (in our case, 1/2 or 0.5), and
- C(n,k) is the binomial coefficient, also known as "n choose k," which represents the number of ways to choose k successes out of n trials.
The binomial coefficient can be calculated using the formula:
C(n, k) = n! / (k! * (n-k)!)
Let's calculate P(8) step by step:
1. Calculate the binomial coefficient:
C(13, 8) = 13! / (8!(13-8)!) = 13! / (8! * 5!) = (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1) = 1287
2. Calculate p^k:
p^k = (1/2)^8 = 1/256 = 0.00390625
3. Calculate (1-p)^(n-k):
(1-p)^(n-k) = (1 - 1/2)^(13-8) = (1/2)^5 = 1/32 = 0.03125
4. Calculate P(8) using the binomial probability formula:
P(8) = C(13, 8) * (1/2)^8 * (1/2)^5 = 1287 * 0.00390625 * 0.03125 ≈ 0.12109375
So, the probability of getting exactly 8 questions correct out of 13 when guessing randomly is approximately 0.1211, or 12.11%.