Suppose Cindy is taking a multiple-choice test where there are 5 possible answers on
each question. If she guesses on all 8 questions, what is the probability that she gets:
a.) exactly 3 correct.
b.) at least 1 correct.
c.) What is the mean?
d.) What is the standard deviation?
To find the probability for each scenario, we need to know the total number of possible outcomes for each question, as well as the number of desired outcomes.
a.) To find the probability that Cindy gets exactly 3 correct answers, we need to calculate the probability of getting 3 correct out of 8 questions. We can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
n is the number of trials (questions) = 8
k is the number of successes (correct answers) = 3
p is the probability of success (getting a correct answer) = 1/5 = 0.2
(1 - p) is the probability of failure (getting an incorrect answer) = 1 - 0.2 = 0.8
Using this formula, we can calculate the probability:
P(X = 3) = (8 choose 3) * (0.2)^3 * (0.8)^(8 - 3)
b.) To find the probability that Cindy gets at least 1 correct answer, we need to calculate the probability of getting 1, 2, 3, 4, 5, 6, 7, or 8 correct answers. We can calculate this by finding the complementary probability and subtracting it from 1:
P(X ≥ 1) = 1 - P(X = 0)
To find P(X = 0), we can plug the values into the binomial probability formula:
P(X = 0) = (8 choose 0) * (0.2)^0 * (0.8)^(8 - 0)
Then, we can calculate P(X ≥ 1) using the formula above.
c.) To find the mean, we can use the formula for the expected value of a binomial distribution:
Mean = n * p
Where n is the number of trials (questions) and p is the probability of success (getting a correct answer).
d.) To find the standard deviation, we can use the formula:
Standard Deviation = sqrt(n * p * (1 - p))
Where n is the number of trials (questions) and p is the probability of success (getting a correct answer).
To answer these questions, we need to understand the probability of getting a correct answer on a single question. Since each question has 5 possible answers and Cindy is guessing randomly, the probability of getting a correct answer on a single question is 1/5 or 0.2.
a.) To find the probability that Cindy gets exactly 3 correct answers, we can use the binomial probability formula:
P(X = k) = (n C k) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting exactly k successes (correct answers)
(n C k) is the number of ways to choose k from n (n choose k)
p is the probability of getting a success on a single trial (probability of getting a correct answer)
1-p is the probability of getting a failure on a single trial (probability of getting a wrong answer)
n is the total number of trials (number of questions)
Plugging in the values:
P(X = 3) = (8 C 3) * (0.2)^3 * (0.8)^(8-3)
Using a combination formula (n C k) = n! / (k! * (n-k)!):
P(X = 3) = (8! / (3! * (8-3)!)) * (0.2)^3 * (0.8)^(8-3)
P(X = 3) = (8! / (3! * 5!)) * (0.2)^3 * (0.8)^5
Calculating this expression will yield the probability that Cindy gets exactly 3 correct answers.
b.) To find the probability that Cindy gets at least 1 correct answer, we can use the complement rule. The probability of getting at least 1 correct answer is equal to 1 minus the probability of getting no correct answers.
P(at least 1 correct) = 1 - P(no correct)
= 1 - (0.8)^8
Calculating this expression will yield the probability that Cindy gets at least 1 correct answer.
c.) The mean (also known as the expected value) refers to the average outcome. In the context of this problem, it represents the average number of correct answers Cindy can expect to get. The mean can be calculated using the formula:
mean = n * p
Plugging in the values:
mean = 8 * 0.2
Calculating this expression will give us the mean.
d.) The standard deviation measures the amount of dispersion or variability in the data. In the context of this problem, it gives an idea of how much Cindy's actual score might deviate from the mean score. The standard deviation of a binomial distribution can be calculated using the formula:
standard deviation = sqrt(n * p * (1-p))
Plugging in the values:
standard deviation = sqrt(8 * 0.2 * (1-0.2))
Calculating this expression will give us the standard deviation.