1. Suppose that you are taking a quiz of 6 multiple- choice questions (the instructor chose the questions randomly), each question having 6 possible responses. You did not study at all for the quiz and will randomly guess the correct response for each question. The random variable X is the number of correct responses.
State the values of n and p (±0.0001):
n =
p =
Calculate the probability (±0.0001) that you will pass this quiz by correctly responding to at least 4 of the 6 questions:
Find the probability (±0.0001) that you will not pass the quiz :
The probability of getting any one question right = 1/6.
"at least 4 of the 6" means 4, 5 or 6 correct.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
P(4) = 1/6^4 * 5/6^2 = ?
P(5) = ?
P(6) = ?
Either-or probabilities are found by adding the individual probabilities.
To find the values of n and p for this problem, let's analyze the situation.
In this case, there are 6 multiple-choice questions, and for each question, there are 6 possible responses. Since you are randomly guessing, each response has an equal chance of being correct. Therefore, the probability of guessing each question correctly is 1/6 or approximately 0.1667.
Now, let's find the values of n and p.
n represents the number of trials or questions in this case, which is 6.
p represents the probability of success for each trial, which is the probability of guessing a question correctly, 0.1667.
Therefore,
n = 6
p = 0.1667 (approximately)
Now, let's calculate the probability of passing the quiz by correctly responding to at least 4 out of 6 questions.
To find this probability, we need to consider the binomial distribution formula:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) represents the probability of getting exactly k correct responses.
(n C k) represents the number of combinations of n trials taken k at a time.
p^k represents the probability of getting k successes.
(1 - p)^(n - k) represents the probability of getting (n - k) failures.
To calculate the probability of passing the quiz by correctly guessing at least 4 questions, we need to calculate the sum of the probabilities of getting exactly 4, 5, and 6 correct responses.
P(X >= 4) = P(X = 4) + P(X = 5) + P(X = 6)
Let's calculate this probability.
P(X = 4) = (6 C 4) * (0.1667)^4 * (1 - 0.1667)^(6 - 4)
P(X = 5) = (6 C 5) * (0.1667)^5 * (1 - 0.1667)^(6 - 5)
P(X = 6) = (6 C 6) * (0.1667)^6 * (1 - 0.1667)^(6 - 6)
Summing up these probabilities will give us the desired probability.
Now, let's calculate the probability that you will not pass the quiz.
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
For this, we need to calculate each individual probability and sum them up.
Let's calculate these probabilities and find the results.