A multiple choice test has five choices. Therefore the probability of getting the correct answer by guessing is 0.2. The test has six questions. Use the binomial formula with n = 6 and p = 0.2 to (a) Determine probability of getting two questions correct.

(b)Determine probability of getting between one and four (inclusive) correct

If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

2 correct = .2^2 * .8^4 = ?

1 correct = .2 * .8^5 = ?

3 correct = .2^3 * .8^3 = ?

4 correct = .2^4 * .8^2 = ?

Either-or probabilities are found by adding the individual probabilities, so add the probabilities you have found.

A unbiased die is thrown seven times. Find the probability of throwing at least five six's.

To determine the probability of getting a specific number of questions correct on a multiple choice test, we can use the binomial formula. The binomial formula gives us the probability of getting a certain number of successes (correct answers) in a fixed number of independent trials (each question).

The binomial formula is:
P(x) = C(n,x) * p^x * (1-p)^(n-x)

Where:
P(x) is the probability of getting x successes
n is the number of trials
x is the number of successes
p is the probability of success on a single trial
C(n,x) is the combination formula, which calculates the number of ways to choose x items from a set of n items.

(a) Determine the probability of getting two questions correct:
Using the binomial formula with n = 6 and p = 0.2, we want to find P(2).
P(2) = C(6,2) * (0.2)^2 * (1-0.2)^(6-2)

Now we need to calculate the combination C(6,2):
C(6,2) = 6! / (2! * (6-2)!)
= 6! / (2! * 4!)
= (6 * 5 * 4!) / (2! * 4!)
= (6 * 5) / 2!
= 30 / 2
= 15

Substituting this value back into the formula:
P(2) = 15 * (0.2)^2 * (1-0.2)^(6-2)
= 15 * 0.04 * 0.8^4
= 0.0576

Therefore, the probability of getting two questions correct is 0.0576.

(b) Determine the probability of getting between one and four (inclusive) correct:
To find the probability of getting between one and four questions correct, we need to sum up the individual probabilities of getting one, two, three, and four correct.

P(1) = C(6,1) * (0.2)^1 * (1-0.2)^(6-1)
P(2) = C(6,2) * (0.2)^2 * (1-0.2)^(6-2)
P(3) = C(6,3) * (0.2)^3 * (1-0.2)^(6-3)
P(4) = C(6,4) * (0.2)^4 * (1-0.2)^(6-4)

Calculating each individual probability using the combination formula, and substituting the values back into the formula:
P(between one and four) = P(1) + P(2) + P(3) + P(4)

You can solve each of the individual probabilities using the same process explained in part (a), and then add them together to get the total probability.