Beth Dahlke is taking a ten question multiple choice test for which each question has three answer choice only one of which is correct. Beth decides on answers by rolling a fair die and making the first answer choice if the die shows 1 or 2 the second if it showes 3 or 4 and the third if it showes 5 or 6. Find the probability of each event a) exactly fseven correct answers b) at least seven correct answers.

Prob of 1 or 2 on a die = 2/6 = 1/3

prob of 2 or 4 = 1/3
prob of 5 or 6 = 1/3

Since these become your choice of answer
prob of selecting 1st answer = 1/3
prob of selecting 2nd answer = 1/3
.....

so the prob of choosing the correct answer in each event is simply 1/3

so to get 7 out of 10 correct
= C(10,7) (1/3)^7 (2/3)^3 = appr .01626

so at least 7
= C(10,7) (1/3)^7 (2/3)^3 + C(10,8) (1/3)^8 (2/3)^2 + .. + C(10,10) (1/3)^10

I will let you do the button pushing

To find the probability of each event, we need to calculate the probability of getting exactly seven correct answers and the probability of getting at least seven correct answers.

a) To find the probability of getting exactly seven correct answers, we can use the binomial probability formula. The probability of getting exactly k successes in n trials, where the probability of success in each trial is p, is given by:

P(X=k) = (nCk) * p^k * (1-p)^(n-k)

In this case, n = 10 (the number of questions), k = 7 (the number of correct answers), and p = 1/3 (the probability of guessing the correct answer).

Using the formula, we can calculate the probability:

P(X=7) = (10C7) * (1/3)^7 * (2/3)^(10-7)

P(X=7) = (10C7) * (1/3)^7 * (2/3)^3

nCk = n! / (k!(n-k)!)
10C7 = 10! / (7!(10-7)!)
10C7 = 10! / (7!3!)
10C7 = (10*9*8) / (3*2*1)
10C7 = 120

P(X=7) = 120 * (1/3)^7 * (2/3)^3
P(X=7) = 120 * (1/2187) * (8/27)
P(X=7) = 0.00385802469

Therefore, the probability of getting exactly seven correct answers is approximately 0.003858.

b) To find the probability of getting at least seven correct answers, we need to calculate the probabilities of getting exactly seven, eight, nine, and ten correct answers, and then sum them up.

P(X≥7) = P(X=7) + P(X=8) + P(X=9) + P(X=10)

We already calculated P(X=7), so let's calculate P(X=8), P(X=9), and P(X=10) using the same formula.

P(X=8) = (10C8) * (1/3)^8 * (2/3)^2
P(X=9) = (10C9) * (1/3)^9 * (2/3)^1
P(X=10) = (10C10) * (1/3)^10 * (2/3)^0

Calculating each probability:

10C8 = 45
P(X=8) = 45 * (1/3)^8 * (2/3)^2

10C9 = 10
P(X=9) = 10 * (1/3)^9 * (2/3)^1

10C10 = 1
P(X=10) = 1 * (1/3)^10 * (2/3)^0

Now, we can find P(X≥7) by summing up these probabilities:

P(X≥7) = P(X=7) + P(X=8) + P(X=9) + P(X=10)

P(X≥7) = 0.003858 + (45 * (1/3)^8 * (2/3)^2) + (10 * (1/3)^9 * (2/3)^1) + (1 * (1/3)^10 * (2/3)^0)

Therefore, the probability of getting at least seven correct answers is the sum of the above terms.

To find the probability of each event, we need to calculate the probability of getting a certain number of correct answers out of ten.

a) To find the probability of exactly seven correct answers, we can use binomial probability. The probability of getting a correct answer is 1/3, and the probability of getting an incorrect answer is 2/3. The number of ways to choose 7 questions out of 10 is given by the combination formula, denoted as C(10, 7). The probability of getting exactly seven correct answers is:

Probability of exactly 7 correct answers = C(10, 7) * (1/3)^7 * (2/3)^(10-7)

Using the combination formula C(n, r) = n! / (r!(n-r)!), where "!" represents factorial, we have:

Probability of exactly 7 correct answers = (10! / (7!(10-7)!)) * (1/3)^7 * (2/3)^3

Simplifying further:

Probability of exactly 7 correct answers = (10*9*8) / (7*6*5) * (1/3)^7 * (2/3)^3

Now, we can calculate this value to get the actual probability.

b) To find the probability of at least seven correct answers, we need to sum up the probabilities of getting exactly 7, 8, 9, and 10 correct answers. We can use the formula:

Probability of at least seven correct answers = Probability of exactly 7 correct answers + Probability of exactly 8 correct answers + Probability of exactly 9 correct answers + Probability of exactly 10 correct answers

Calculate each probability separately using the method described above and then sum them up to get the probability of at least seven correct answers.

Note: The exact calculations are not provided here as they involve further arithmetic operations.