A multiple choice quiz contains five question. Each question has four answer choices. Michael is not prepared for the quiz and decides to guess for each question. What is the probability that Michael will get at least one question correct? What is the probability that Micheal will get all five questions correct? Could you show me how to work it out please.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
"At least one" = one or more.
1/4 * (3/4)^4 = ? (one correct)
(1/4)^2 * (3/4)^3 = ? (2 correct)
(1/4)^3 * (3/4)^2 = ? (3 correct)
(1/4)^4 * 3/4 = ? (4 correct)
(1/4)^5 = ? (5 correct)
Either-or probabilities are found by adding the individual probabilities.
To find the probability that Michael will get at least one question correct, we need to find the probability of two scenarios: the probability of getting one question correct and the probability of getting multiple questions correct.
First, let's calculate the probability of getting one question correct. Since each question has four answer choices and Michael is guessing, the probability of guessing correctly for one question is 1/4. Therefore, the probability of getting one question correct is 1/4.
Now, let's calculate the probability of getting multiple questions correct. Since each question is independent, the probability of getting multiple questions correct is calculated by multiplying the probabilities of getting each question correct. In this case, we want to calculate the probability of getting all five questions correct.
The probability of getting all five questions correct can be found by multiplying the probability of getting one question correct (1/4) by itself five times (since there are five questions).
Therefore, the probability of getting all five questions correct is (1/4)^5 = 1/1024.
Now, to find the probability of getting at least one question correct, we can subtract the probability of getting no questions correct from 1.
Probability of getting no questions correct = Probability of getting all questions incorrect = 1 - Probability of getting at least one question correct
Since there are only two possibilities (getting at least one question correct or getting no questions correct), we can use this formula to find the desired probability.
Probability of getting at least one question correct = 1 - Probability of getting no questions correct
To calculate the probability of getting no questions correct, we can use the same logic as before, but with the probability of guessing each question incorrectly (3/4).
Therefore, the probability of getting no questions correct is (3/4)^5 = 243/1024.
Substituting this value into our formula, the probability of getting at least one question correct is:
Probability of getting at least one question correct = 1 - 243/1024.
By simplifying the equation, we get the final answer:
Probability of getting at least one question correct = 781/1024.
So, the probability that Michael will get at least one question correct is 781/1024. And the probability that Michael will get all five questions correct is 1/1024.
To find the probability that Michael will get at least one question correct, we need to find the probability that he gets none of the questions correct and then subtract it from 1.
The probability of choosing the wrong answer for each question is 3/4 since there are four answer choices and only one is correct.
So, the probability of getting none of the questions correct is (3/4)^5.
Therefore, the probability that Michael will get at least one question correct is 1 - (3/4)^5 ≈ 0.7627, or 76.27%.
To find the probability that Michael will get all five questions correct, we need to multiply the probability of choosing the correct answer for each question.
The probability of choosing the correct answer for each question is 1/4.
So, the probability of getting all five questions correct is (1/4)^5 = 1/1024 ≈ 0.0009766, or 0.09766%.
I hope this clarifies the calculation for you! Let me know if you have any other questions.