On a very short quiz, there is one multiple choice question with 5 possible choices (a, b, c, d, e) and one true or false question. Assume you are taking the quiz but do not have any idea what the correct answer is to either question. You mark an answer to each question anyway. Assume that the events are independent of each other.

A) What is the probability that you have given the correct answer to both questions?

B) What is the probability that only one of the two answers is correct?

C) What is the probability that neither answer is correct?

D) What is the probability that only your answer to the multiple choice question is correct?

E) What is the probability that you have answered only the true or false question correctly?

Probability of all/both events occurring is found by multiplying the individual probabilities.

A) 1/5 * 1/2 = ?

Probability of either/or events occurring is found by adding the individual probabilities.

B) Follow above instructions.

c) (1-1/5)(1-1/2) = ?

This info should lead you to figuring out D and E.

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Let's solve each question step-by-step:

A) The probability of giving the correct answer to the multiple choice question is 1/5, since there are 5 possible choices and only one correct answer. The probability of giving the correct answer to the true or false question is 1/2, since there are 2 possible choices (true or false) and only one correct answer. Since the events are independent, we can multiply the probabilities: (1/5) * (1/2) = 1/10.

B) To find the probability that only one of the two answers is correct, we need to consider two cases:
- Case 1: The multiple choice question is correct and the true or false question is not.
The probability of this case is (1/5) * (1/2) = 1/10.
- Case 2: The multiple choice question is not correct and the true or false question is correct.
The probability of this case is (4/5) * (1/2) = 2/10.

To find the total probability, we add the probabilities of the two cases: 1/10 + 2/10 = 3/10.

C) The probability that neither answer is correct can be found by subtracting the probability of getting either the multiple choice question or the true or false question correct from 1.
- The probability of giving the correct answer to the multiple choice question is 1/5.
- The probability of giving the correct answer to the true or false question is 1/2.
Thus, the probability of either one of the two questions being correct is 1/5 + 1/2 = 7/10.
Therefore, the probability that neither answer is correct is 1 - 7/10 = 3/10.

D) The probability of giving the correct answer to the multiple choice question is 1/5. The probability of giving the incorrect answer to the true or false question is also 1/2. Therefore, the probability that only your answer to the multiple choice question is correct is (1/5) * (1/2) = 1/10.

E) The probability of giving the incorrect answer to the multiple choice question is 4/5. The probability of giving the correct answer to the true or false question is 1/2. Therefore, the probability that you have answered only the true or false question correctly is (4/5) * (1/2) = 2/5.

I hope this helps! Let me know if you have any further questions.

To find the probabilities for each scenario, we can use the principle of independent events, which states that the probability of two independent events occurring together is the product of their individual probabilities.

Let's analyze each question separately:

Question 1: Multiple choice question
- Number of possible choices: 5 (a, b, c, d, e)
- Probability of selecting the correct choice: 1/5 (since there is only one correct choice out of 5 options)

Question 2: True or false question
- Number of possible choices: 2 (true or false)
- Probability of selecting the correct choice: 1/2 (since there is only one correct choice out of 2 options)

A) Probability of giving the correct answer to both questions:
Since the events are independent, we can multiply the individual probabilities:
P(A) = (1/5) * (1/2) = 1/10 = 0.1

B) Probability of only one of the two answers being correct:
There are two possibilities for this scenario:
i) Correct answer to the multiple choice question and wrong answer to the true or false question.
ii) Wrong answer to the multiple choice question and correct answer to the true or false question.

P(B) = (1/5) * (1/2) + (4/5) * (1/2) = 1/10 + 2/5 = 3/10 = 0.3

C) Probability of neither answer being correct:
In this case, we need to calculate the probability of giving a wrong answer to both questions.

P(C) = (4/5) * (1/2) = 2/5 = 0.4

D) Probability of only the multiple choice question answer being correct:
In this case, the answer to the true or false question is incorrect.

P(D) = (1/5) * (1/2) = 1/10 = 0.1

E) Probability of only the true or false question answer being correct:
Here, the answer to the multiple-choice question is incorrect.

P(E) = (4/5) * (1/2) = 2/5 = 0.4

So, the probabilities for each scenario are:
A) 0.1
B) 0.3
C) 0.4
D) 0.1
E) 0.4