a test consists of 25 multiple choice questions. Each has 5 possible answers, which only one is correct. If a student guesses on each question, find the following:
a) prob that he will guess all of them right
b)prob that he will guess at most 12 right
c) prob that he will guess at least 1 right
d) mean and stndrd. deviation of the number of correct answers
e)estimate the prob of the number of correct answers that will fall within the limits -----> miu +or- 2(stand.dev)
To solve these problems, we can use the binomial probability formula since each question has two possible outcomes (correct or incorrect) and the probability of either outcome is the same for each question. The formula is as follows:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of getting exactly k correct answers
- n is the number of trials (questions = 25 in this case)
- k is the number of successful outcomes (correct answers)
- nCk is the combination formula (n choose k)
- p is the probability of a successful outcome (correct answer = 1/5 = 0.2)
- (1-p) is the probability of an unsuccessful outcome (incorrect answer = 1 - 0.2 = 0.8)
Let's solve each part of the problem step by step:
a) Probability of getting all 25 questions right:
P(X=25) = (25C25) * (0.2)^25 * (1-0.2)^(25-25)
Calculating the values:
P(X=25) = 1 * 0.2^25 * 0.8^0
P(X=25) = 0.2^25 ≈ 0.0000000008
So, the probability that the student guesses all of them right is approximately 0.0000000008.
b) Probability of guessing at most 12 right:
We need to calculate the probability of getting 0, 1, 2, ..., 12 correct answers and add them together.
P(X ≤ 12) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=12)
P(X ≤ 12) = ∑ (25Ck) * (0.2)^k * (0.8)^(25-k) for k = 0 to 12
Using a calculator or statistical software can help with this calculation.
c) Probability of guessing at least 1 right:
To find the probability of guessing at least 1 right, we can subtract the probability of guessing none right from 1.
P(X ≥ 1) = 1 - P(X=0)
d) Mean and standard deviation of the number of correct answers:
The mean (μ) of a binomial distribution is given by μ = n * p.
The standard deviation (σ) of a binomial distribution is given by σ = √(n * p * (1-p)).
Substituting the values into the formulas:
μ = 25 * 0.2 = 5
σ = √(25 * 0.2 * 0.8) ≈ 2
So, the mean (average) number of correct answers is 5, and the standard deviation is approximately 2.
e) Estimating the probability of the number of correct answers falling within μ ± 2σ:
We need to calculate the probability of getting 5 - 2 = 3, 4, 5, 6, and 7 correct answers and add them together.
P(3 ≤ X ≤ 7) = P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7)
P(3 ≤ X ≤ 7) = ∑ (25Ck) * (0.2)^k * (0.8)^(25-k) for k = 3 to 7
Using a calculator or statistical software can help with this calculation.
I hope this explanation helps you understand the process of solving the problems using the binomial probability formula.