A multiple choice test has 16 questions. Each question has 4 possible answers with
only one is correct. Assume that all questions are answered and the probability to
answer any question correctly is 0.25. Let X be the number of questions answered
correctly among 16.
1. What model would you propose for X ? Compute E(X) and V (X).
2. What is the probability that at least 4 questions are answered correctly in the
test ?
3. What is the probability that 7, 8, 9, or 10 questions are answered correctly ?
Sarah got 80% on her math quiz. If she answered 12 questions correctly, how many question were on the test?
Fhshnanxnsan
1. The model that can be proposed for X is a binomial distribution since each question has only two outcomes - answering correctly or incorrectly - and the probability of success (answering correctly) is constant at 0.25.
The binomial distribution can be described as X ~ B(n,p), where n is the number of trials (in this case, 16 questions) and p is the probability of success (answering correctly, 0.25).
To compute E(X) (expected value) and V(X) (variance), we can use the formulas for the binomial distribution:
E(X) = n * p
E(X) = 16 * 0.25
E(X) = 4
V(X) = n * p * (1 - p)
V(X) = 16 * 0.25 * (1 - 0.25)
V(X) = 3
Therefore, the expected value of X is 4 and the variance of X is 3.
2. To find the probability that at least 4 questions are answered correctly, we need to calculate the sum of probabilities for X ≥ 4. This can be done by subtracting the sum of probabilities for X ≤ 3 from 1.
P(X ≥ 4) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]
Using the binomial distribution formula, we can calculate the probabilities:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
P(X = 0) = (16 choose 0) * 0.25^0 * (1-0.25)^(16-0) = 0.035
P(X = 1) = (16 choose 1) * 0.25^1 * (1-0.25)^(16-1) = 0.147
P(X = 2) = (16 choose 2) * 0.25^2 * (1-0.25)^(16-2) = 0.294
P(X = 3) = (16 choose 3) * 0.25^3 * (1-0.25)^(16-3) = 0.324
Now we can calculate the probability:
P(X ≥ 4) = 1 - (0.035 + 0.147 + 0.294 + 0.324) = 1 - 0.8 = 0.2
Therefore, the probability that at least 4 questions are answered correctly is 0.2.
3. To find the probability that 7, 8, 9, or 10 questions are answered correctly, we need to calculate the sum of probabilities for X = 7, X = 8, X = 9, and X = 10.
P(X = 7) = (16 choose 7) * 0.25^7 * (1-0.25)^(16-7)
P(X = 8) = (16 choose 8) * 0.25^8 * (1-0.25)^(16-8)
P(X = 9) = (16 choose 9) * 0.25^9 * (1-0.25)^(16-9)
P(X = 10) = (16 choose 10) * 0.25^10 * (1-0.25)^(16-10)
Now we can calculate the probability:
P(X = 7) = 0.309
P(X = 8) = 0.233
P(X = 9) = 0.131
P(X = 10) = 0.054
P(X = 7, 8, 9, or 10) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) = 0.309 + 0.233 + 0.131 + 0.054 = 0.727
Therefore, the probability that 7, 8, 9, or 10 questions are answered correctly is 0.727.
To solve these questions, we can use the binomial distribution. The binomial distribution is used to model the number of successes (in this case, correctly answered questions) in a fixed number of independent Bernoulli trials (each question answered has a 0.25 probability of being correct).
1. The model for X is a binomial distribution with parameters n = 16 (the number of trials/questions) and p = 0.25 (the probability of success in each trial/question). The expected value of X (E(X)) can be calculated using the formula E(X) = n * p. Therefore, E(X) = 16 * 0.25 = 4.
To calculate the variance of X (V(X)), we use the formula V(X) = n * p * (1 - p). Therefore, V(X) = 16 * 0.25 * (1 - 0.25) = 3.
2. To calculate the probability that at least 4 questions are answered correctly, we need to sum the probabilities of 4, 5, 6, ..., 16 questions being answered correctly. Using the binomial distribution formula, we can calculate each probability and sum them up.
P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) + ... + P(X = 16)
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
C(n, k) represents the number of ways to choose k successes from n trials, and can be calculated as C(n, k) = n! / (k! * (n - k)!)
Using this formula, we can calculate each probability and sum them up to get the final answer.
3. To calculate the probability that 7, 8, 9, or 10 questions are answered correctly, we can sum the probabilities of each of these individual events occurring.
P(7 ≤ X ≤ 10) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
Again, using the binomial distribution formula, we can calculate each probability and sum them up to get the final answer.
Keep in mind that these calculations assume the questions are independent and have the same probability of being answered correctly.