I have no idea where to start!
*A quick quiz consists of 3 multiple choice problems, each of which has 4 answers, only one of which is correct. If you make random guesses on all 3 problems,
(a) What is the probability that all 3 of your answers are incorrect?
(b) What is the probability that all 3 of your answers are correct?
a) The prob that a guess is wrong is 3/4
so the prob that none are correct, that is, all 3 are wrong is (3/4)(3/4)(3/4) = 27/64.
b) then the prob that all 3 are guessed correctly is (1/4)(1/4)(1/4) = 1/64
Okay great thank you so much that makes much more sense!
To find the probabilities for (a) and (b), we can use the concept of independent events. Each problem is a separate event, and the probability of a correct or incorrect answer remains the same for each problem.
Let's calculate the probabilities step-by-step:
(a) Probability that all 3 answers are incorrect:
Since there are 4 possible answers for each problem and only one is correct, the probability of getting a single problem incorrect is 3/4 (or 0.75). Since the answers are independent, we can multiply the probabilities together:
P(incorrect answer) = 3/4
P(incorrect answer) x P(incorrect answer) x P(incorrect answer) = (3/4) x (3/4) x (3/4) = 27/64 ≈ 0.4219
So, the probability that all 3 answers are incorrect is approximately 0.4219 or 42.19%.
(b) Probability that all 3 answers are correct:
Similarly, the probability of getting a single problem correct is 1/4 (or 0.25). Since the answers are independent, we can multiply the probabilities together:
P(correct answer) = 1/4
P(correct answer) x P(correct answer) x P(correct answer) = (1/4) x (1/4) x (1/4) = 1/64 ≈ 0.0156
So, the probability that all 3 answers are correct is approximately 0.0156 or 1.56%.
To calculate the probability of a specific event happening, we need to know the total number of possible outcomes and the number of favorable outcomes.
In this case, each multiple choice problem has 4 answers, of which only 1 is correct. If you make random guesses on all 3 problems, you have a 1/4 chance of getting each question correct.
(a) The probability of getting a specific question incorrect is 3/4 since there are 3 incorrect answers out of 4 possible answers. Since each question is independent, the probability of getting all 3 answers incorrect is (3/4) * (3/4) * (3/4) = 27/64, which is approximately 0.422 or 42.2%.
(b) The probability of getting a specific question correct is 1/4. Since each question is independent, the probability of getting all 3 answers correct is (1/4) * (1/4) * (1/4) = 1/64, which is approximately 0.016 or 1.6%.
To summarize:
(a) The probability that all 3 answers are incorrect is 27/64 or approximately 0.422.
(b) The probability that all 3 answers are correct is 1/64 or approximately 0.016.