1. Testing a pain reliever using 840 people to determine if it is effective. The random variable represents the number of people who find the pain reliever to be effective.
a. binomial experiment
b. not a binomial experiment
A test consists of 10 true or false questions. To pass the test a student must answer at least eight questions correctly. If the student guesses on each question, what is the probability that the student will pass the test?
a. 0.20
b. 0.055
c. 0.08
d. 0.8
1. a?
To have 8 or more correct, the student must get exactly 8 correct OR exactly 9 correct OR exactly 10 correct
= C(10,8)(1/2)^8 (1/2)^2 + C(10,9) (1/2)^10 + C(10,10)(1/2)^10
= (1/2)^10(45 + 10 + 1)
= 56/1024
= 7/128
1. To determine if the testing of a pain reliever using 840 people is a binomial experiment or not, we need to check if it satisfies the four conditions for a binomial experiment:
- The experiment consists of a fixed number of trials: Yes, since there are 840 people involved; thus, the number of trials is fixed.
- Each trial has two possible outcomes: Yes, either a person finds the pain reliever effective or not.
- The outcomes of each trial are independent of each other: Yes, each person's response is independent of others.
- The probability of success (finding the pain reliever effective) remains constant: Yes, as long as we assume the pain reliever's effectiveness remains consistent.
Since all four conditions are satisfied, the experiment is a binomial experiment.
2. To find the probability that the student will pass the test, we can use the binomial probability formula. Let's break down the problem:
- Number of trials (n): 10 (as there are 10 questions)
- Probability of success (p): 0.5 (since it's a guess between true or false)
- Number of successes required (r): At least 8 out of 10 questions correct
To find the probability of passing the test (at least 8 correct), we need to sum the probabilities of getting exactly 8, 9, and 10 questions correct.
P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10)
Using the binomial probability formula:
P(X = k) = nCk * p^k * (1-p)^(n-k)
Plugging in the values:
P(X = 8) = 10C8 * (0.5)^8 * (0.5)^(10-8)
P(X = 9) = 10C9 * (0.5)^9 * (0.5)^(10-9)
P(X = 10) = 10C10 * (0.5)^10 * (0.5)^(10-10)
Finally, add these probabilities together.
Calculating the probabilities and adding them up, we find that the probability of passing the test (at least 8 correct) is approximately 0.1094.
None of the provided options matches this probability, so none of the provided choices (a, b, c, or d) are correct.