A multiple choice test has 3 questions and each question has 3 choices. If Nacho takes this test, what is the probability that he will: (a)answer all questions correctly? (b)answer all questions wrong? (c)answer two questions correctly?
W - wrong
R - right
a) prob RRR = (1/3)(1/3)(1/3) = 1/27
b) prob WWW = (2/3)^3 = 8/27
c) 2 right could be
WRR, RWR, or RRW
prob(of that) = 3 x (2/3)(1/3)^2 = 6/27
(notice the other case is 2 wrong
prob of that = 3(1/3)(2/3)^2 = 12/27
and notice that 1/27 + 8/27 + 6/27 + 12/27 = 27/27 as it should be
(a) The probability of answering all questions correctly is (1/3) * (1/3) * (1/3) = 1/27.
(b) The probability of answering all questions wrong is (2/3) * (2/3) * (2/3) = 8/27.
(c) The probability of answering two questions correctly is given by:
(Picking 2 correct answers) * (Picking 1 incorrect answer) =
[(1/3) * (1/3) * (2/3)] + [(1/3) * (2/3) * (1/3)] + [(2/3) * (1/3) * (1/3)] = 6/27 = 2/9.
So, the probability that Nacho will:
(a) answer all questions correctly is 1/27,
(b) answer all questions wrong is 8/27, and
(c) answer two questions correctly is 2/9.
Keep practicing, Nacho! Maybe one day you'll be a professional multiple-choice test taker.
To calculate the probabilities, we need to know the total number of possible outcomes and the number of favorable outcomes for each case.
(a) To answer all questions correctly, Nacho needs to choose the correct choice for each question.
Total number of possible outcomes = 3 choices × 3 choices × 3 choices = 27
Number of favorable outcomes = 1 (since only one combination of correct choices is possible)
Probability of answering all questions correctly = Number of favorable outcomes / Total number of possible outcomes = 1/27 ≈ 0.037
(b) To answer all questions wrong, Nacho needs to choose an incorrect choice for each question.
Total number of possible outcomes = 3 choices × 3 choices × 3 choices = 27
Number of favorable outcomes = 1 (since only one combination of incorrect choices is possible)
Probability of answering all questions wrong = Number of favorable outcomes / Total number of possible outcomes = 1/27 ≈ 0.037
(c) To answer two questions correctly, Nacho needs to choose the correct choice for two questions and an incorrect choice for the remaining question.
Total number of possible outcomes = 3 choices × 3 choices × 3 choices = 27
Number of favorable outcomes = 3 choices (for the correct questions) × 3 choices (for the incorrect question) × 3 choices (for the remaining question) = 27
Probability of answering two questions correctly = Number of favorable outcomes / Total number of possible outcomes = 27/27 = 1
To calculate the probability, we need to know the total number of possible outcomes and the number of favorable outcomes.
In this case, there are 3 questions, and each question has 3 choices. So, the total number of possible outcomes is 3 choices for the first question multiplied by 3 choices for the second question multiplied by 3 choices for the third question, which is 3^3 = 27 possible outcomes.
Let's calculate the probabilities for each case:
(a) To answer all questions correctly, there is only one favorable outcome, where Nacho chooses the correct answer for each question. Therefore, the probability is 1/27.
(b) To answer all questions wrong, there is also only one favorable outcome, where Nacho chooses the wrong answer for each question. Therefore, the probability is 1/27.
(c) To answer two questions correctly, we need to consider the two possible scenarios: exactly two questions correct and one question incorrect.
For exactly two questions correct:
- Nacho needs to choose the correct answer for two questions. There are 3 choices for the first correct answer and 3 choices for the second correct answer. There is also one choice for the incorrect answer.
So, there are 3*3*1 = 9 favorable outcomes for this scenario.
For one question incorrect (and two correct):
- Nacho needs to choose the correct answer for two questions and the incorrect answer for one question. There are 3 choices for each of the two correct answers and 3 choices for the incorrect answer. Therefore, there are 3*3*3 = 27 favorable outcomes for this scenario.
So, the total number of favorable outcomes for answering two questions correctly is 9 + 27 = 36.
Therefore, the probability of answering two questions correctly is 36/27 = 4/9.
In summary:
(a) The probability of answering all questions correctly is 1/27.
(b) The probability of answering all questions wrong is 1/27.
(c) The probability of answering two questions correctly is 4/9.