A multiple-choice test has 20 questions with 4 answer choices. If a student guesses on every question, what is the probability of getting 8 questions correct? (hint: use the binomial probability formula)

The binomial probability formula is P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k)), where:

• P(X=k) is the probability of k successes in n trials
• C(n, k) is the combination of n items taken k at a time
• p is the probability of success on a single trial
• n is the number of trials

In this case, the student is guessing on every question, so the probability of success on a single trial (p) is 1/4 (since there are 4 answer choices and only 1 is correct). The number of trials (n) is 20 (since there are 20 questions), and we want to find the probability of getting 8 questions correct (k=8).

First, calculate C(n, k) = C(20, 8). This is the number of ways to choose 8 questions to get correct out of 20.

C(n, k) = n! / [k!(n-k)!] = 20! / [8!(20-8)!] = 125970

Next, calculate p^k = (1/4)^8 = 0.00001526

Then, calculate (1-p)^(n-k) = (3/4)^(20-8) = 0.03167629

Finally, multiply these three values together to get the probability:

P(X=8) = C(n, k) * (p^k) * ((1-p)^(n-k)) = 125970 * 0.00001526 * 0.03167629 = 0.0609

So, the probability of a student guessing and getting exactly 8 questions correct on a 20-question multiple-choice test (with 4 answer choices per question) is approximately 0.0609, or 6.09%.

The binomial probability formula is P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k)), where:

• P(X=k) is the probability of k successes in n trials
• C(n, k) is the combination of n items taken k at a time
• p is the probability of success on a single trial
• n is the number of trials
• k is the number of successes

In this case, the student is guessing on every question, so the probability of success on a single trial (p) is 1/4 (since there are 4 answer choices and only 1 is correct). The number of trials (n) is 20 (the number of questions), and the number of successes (k) we're interested in is 8 (the number of questions the student gets right).

So, we can plug these values into the formula:

P(X=8) = C(20, 8) * ((1/4)^8) * ((1 - 1/4)^(20 - 8))

First, calculate C(20, 8), which is the number of ways to choose 8 questions out of 20. This is calculated as 20! / (8!(20 - 8)!), which equals 125970.

Next, calculate (1/4)^8, which is the probability of guessing 8 questions correctly. This equals 0.00001526.

Then, calculate (1 - 1/4)^(20 - 8), which is the probability of guessing the remaining 12 questions incorrectly. This equals 0.01274.

Finally, multiply these three values together to get the probability:

P(X=8) = 125970 * 0.00001526 * 0.01274 = 0.0244 or 2.44%.

So, if a student guesses on every question, the probability of getting 8 questions correct is approximately 2.44%.

The binomial probability formula is P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k)), where:

• P(X=k) is the probability of k successes in n trials
• C(n, k) is the combination of n items taken k at a time
• p is the probability of success on a single trial
• n is the number of trials
• k is the number of successes

In this case, the student is guessing on every question, so p = 1/4 (since there are 4 answer choices and only 1 is correct). The number of trials, n, is 20 (the number of questions), and the number of successes, k, is 8 (the number of questions the student wants to get right).

So, we can plug these values into the formula:

P(X=8) = C(20, 8) * ((1/4)^8) * ((1 - 1/4)^(20-8))

First, calculate C(20, 8), which is the number of ways to choose 8 questions out of 20. This is calculated as 20! / (8!(20-8)!), which equals 125,970.

Next, calculate (1/4)^8, which is the probability of guessing 8 questions correctly. This equals approximately 0.00001526.

Then, calculate (1 - 1/4)^(20-8), which is the probability of guessing the remaining 12 questions incorrectly. This equals approximately 0.031676.

Finally, multiply these three values together to get the probability:

P(X=8) = 125,970 * 0.00001526 * 0.031676 = approximately 0.0608, or 6.08%.

So, if a student guesses on every question, the probability of getting 8 questions correct is approximately 6.08%.

The binomial probability formula is P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k)), where:

• P(X=k) is the probability of k successes in n trials
• C(n, k) is the combination of n items taken k at a time
• p is the probability of success on a single trial
• n is the number of trials
• k is the number of successes

In this case, the student is guessing on every question, so the probability of success on a single trial (p) is 1/4 (since there are 4 answer choices and only 1 is correct). The number of trials (n) is 20 (since there are 20 questions), and the number of successes (k) we're interested in is 8 (since we want to find the probability of getting 8 questions correct).

So, we can plug these values into the binomial probability formula:

P(X=8) = C(20, 8) * ((1/4)^8) * ((1 - 1/4)^(20 - 8))

First, calculate C(20, 8), which is the number of ways to choose 8 questions out of 20. This is calculated as 20! / (8!(20 - 8)!), which equals 125970.

Next, calculate (1/4)^8, which is the probability of guessing 8 questions correctly. This equals 0.00001526.

Then, calculate (1 - 1/4)^(20 - 8), which is the probability of guessing the remaining 12 questions incorrectly. This equals 0.01293945.

Finally, multiply these three values together to get the probability:

P(X=8) = 125970 * 0.00001526 * 0.01293945 = 0.0246

So, the probability of a student guessing and getting 8 questions correct on a 20-question multiple-choice test with 4 answer choices is approximately 0.0246, or 2.46%.

The binomial probability formula is P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k)), where:

• P(X=k) is the probability of k successes in n trials
• C(n, k) is the combination of n items taken k at a time
• p is the probability of success on a single trial
• n is the number of trials
• k is the number of successes

In this case, the student is guessing on every question, so the probability of success on a single trial (p) is 1/4 (since there are 4 answer choices and only 1 is correct). The number of trials (n) is 20 (since there are 20 questions), and the number of successes (k) we're interested in is 8 (since we want to find the probability of getting 8 questions correct).

So, we can plug these values into the binomial probability formula:

P(X=8) = C(20, 8) * ((1/4)^8) * ((1 - 1/4)^(20 - 8))

First, calculate C(20, 8), which is the number of ways to choose 8 questions out of 20. This is calculated as 20! / (8!(20 - 8)!), where "!" denotes factorial.

Then, calculate (1/4)^8, which is the probability of guessing 8 questions correctly.

Next, calculate (1 - 1/4)^(20 - 8), which is the probability of guessing the remaining 12 questions incorrectly.

Finally, multiply these three values together to get the probability of guessing exactly 8 questions correctly on a 20-question multiple-choice test.