A multiple choice test consists of four questions. Each question has five possible answers of which
only one is correct. A student guesses on every question. Find the probability distribution of X, the
number of questions she answers correctly.
A) x P(X = x)
0 0
1 0.2
2 0.04
3 0.008
4 0.0016
B) x P(X = x)
0 0.4096
1 0.4096
2 0.1536
3 0.0256
4 0.0016
C) x P(X = x)
0 0.4096
1 0.4096
2 0.1024
3 0.0768
4 0.0016
D) x P(X = x)
1 0.4096
2 0.1536
3 0.0256
4 0.0016
Church
To find the probability distribution of X, the number of questions answered correctly, we need to consider the probability of getting each possible value of X.
In this case, there are four questions, and for each question, there are five possible answers, only one of which is correct. Since the student is guessing on every question, the probability of guessing the correct answer is 1/5 for each question.
To find the probability of getting a specific number of questions correct, we need to use the binomial probability formula. The formula is:
P(X = x) = (nCx) * p^x * (1-p)^(n-x)
Where:
- n is the number of trials (number of questions in this case)
- x is the number of successful outcomes (number of correct answers)
- p is the probability of success (probability of guessing the correct answer, in this case 1/5 for each question)
- (nCx) is the binomial coefficient, calculated as n! / (x!(n-x)!)
Now, we can calculate the probabilities for each possible value of X:
For x = 0:
P(X = 0) = (4C0) * (1/5)^0 * (4/5)^(4-0)
= 1 * 1 * (4/5)^4
= (4/5)^4
= 0.4096
For x = 1:
P(X = 1) = (4C1) * (1/5)^1 * (4/5)^(4-1)
= 4 * (1/5) * (4/5)^3
= 4 * (4/125)
= 16/125
= 0.128
For x = 2:
P(X = 2) = (4C2) * (1/5)^2 * (4/5)^(4-2)
= 6 * (1/25) * (4/5)^2
= 6 * (4/625)
= 24/625
= 0.0384
For x = 3:
P(X = 3) = (4C3) * (1/5)^3 * (4/5)^(4-3)
= 4 * (1/125) * (4/5)^1
= 4 * (4/125) * (4/5)
= 64/625
= 0.1024
For x = 4:
P(X = 4) = (4C4) * (1/5)^4 * (4/5)^(4-4)
= 1 * (1/625) * 1
= 1/625
= 0.0016
Therefore, the probability distribution of X is:
A) x P(X = x)
0 0.4096
1 0.128
2 0.0384
3 0.1024
4 0.0016
This matches with option C) in the given choices.