Suppose a multiple choice test has 20 questions with each question having 4 choices, only one of which is correct that is %25. suppose an unprepared student writes the answers, each time randomly picking on the 4 choices. what is the probability that the student will get
A) exactly 4 answers correct
B)All the answers incorrect
C) at least 8 correct
D) what is mean and the standard deviation
To find the probabilities for each scenario, we need to use the concept of binomial distribution.
For multiple choice questions with 4 choices (only one of which is correct), the probability of randomly guessing the correct answer is 1/4, which is 25%.
A) Probability of exactly 4 correct answers:
To calculate the probability of getting exactly 4 correct answers, we use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes
- n is the number of trials (number of questions)
- k is the number of successes (number of correct answers)
- p is the probability of success in a single trial (probability of getting a correct answer)
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
In this case:
- n = 20 (number of questions)
- k = 4 (number of correct answers)
- p = 1/4 (probability of getting a correct answer)
Plugging these values into the formula:
P(X = 4) = (20 choose 4) * (1/4)^4 * (3/4)^(20-4)
Using a calculator or software, you can calculate (20 choose 4) to be 4845. Evaluating the rest of the expression:
P(X = 4) ≈ 0.0034
So, the probability that the student will get exactly 4 answers correct is approximately 0.0034.
B) Probability of all incorrect answers:
To find the probability that the student will get all answers incorrect, we can subtract the probability of getting any correct answers from 1.
In this case, the probability of getting a single question incorrect (choosing one of the incorrect options) is 3/4. Since each question is independent, we can multiply this probability for all 20 questions.
P(all incorrect) = (3/4)^20 ≈ 0.0004
So, the probability that the student will get all answers incorrect is approximately 0.0004.
C) Probability of at least 8 correct answers:
To find the probability of getting at least 8 correct answers, we need to calculate the probability of getting 8, 9, 10, ..., up to 20 correct answers and sum them up.
P(at least 8 correct) = P(X = 8) + P(X = 9) + ... + P(X = 20)
Using the same binomial probability formula as in part A, you can calculate the probabilities for each number of correct answers and sum them up.
D) Mean and standard deviation:
The mean and standard deviation for a binomial distribution can be calculated using the following formulas:
Mean (μ) = n * p
Standard Deviation = sqrt(n * p * (1-p))
In this case:
- n = 20 (number of questions)
- p = 1/4 (probability of getting a correct answer)
Plugging these values into the formulas:
Mean (μ) = 20 * (1/4) = 5
Standard Deviation = sqrt(20 * (1/4) * (3/4)) ≈ 1.94
So, the mean number of correct answers for the student is 5, and the standard deviation is approximately 1.94.