Claire takes a five-question true-false test. She hasn't studied, so she guesses at random. With her ESP she figures she has a 60% probability of getting any one answer wrong.

A. What is the probability of getting an answer wrong? (Do you think they want me to ignore the ESP nugget or take it into account?)
b. Let P(x) be the probability of getting exactly x of the five answers right. Calculate p(0) through P(5)
c. What is the probability of getting at least three answers right.

We will have to assume the ESP estimation is correct.

The resulting distribution is binomial, with 5 questions, and probability of success p=1-0.6=0.4, q=1-p=0.6 (failure).

The probability of getting exactly n question correct (out of 5) is given by the binomial coefficient
C(5,n)p^n q^(5-n)
where C(5,n)=5!/(n!(5-n)!)
Example:
probability of getting 3 answers right
=P(3) = [5!/(3!2!)]0.4^3 0.6^2
=0.2304

To calculate the probability of getting at least three correct would be:
P(3)+P(4)+P(5)=0.317

A. To calculate the probability of getting an answer wrong, we need to consider that Claire has a 60% probability of getting any one answer wrong. Therefore, the probability of getting an answer wrong would be 0.6 or 60%.

B. To calculate the probabilities of getting exactly x out of the five answers right, we can use the binomial probability formula:

P(x) = C(n, x) * p^x * (1-p)^(n-x)

Where:
- P(x) is the probability of getting exactly x out of the five answers right.
- C(n, x) is the number of combinations of n items taken x at a time, which can be calculated using the binomial coefficient formula: n! / (x! * (n-x)!)
- p is the probability of getting a single answer right, which is the complement of the probability of getting it wrong. In this case, p = 1 - 0.6 = 0.4.
- n represents the total number of questions, which is 5.
- x represents the number of questions answered correctly, ranging from 0 to 5.

We can calculate P(0) through P(5) as follows:

P(0) = C(5, 0) * 0.4^0 * (1 - 0.4)^(5-0)
P(1) = C(5, 1) * 0.4^1 * (1 - 0.4)^(5-1)
P(2) = C(5, 2) * 0.4^2 * (1 - 0.4)^(5-2)
P(3) = C(5, 3) * 0.4^3 * (1 - 0.4)^(5-3)
P(4) = C(5, 4) * 0.4^4 * (1 - 0.4)^(5-4)
P(5) = C(5, 5) * 0.4^5 * (1 - 0.4)^(5-5)

By calculating these values using the given formulas, you can determine the probabilities of getting exactly x out of the five answers right.

C. To calculate the probability of getting at least three answers right, we need to sum up the probabilities of getting exactly 3, 4, or 5 answers right.

P(at least 3) = P(3) + P(4) + P(5)

By substituting the values calculated in step B, you can determine the probability of getting at least three answers right.