A multiple choice test consists of eight questions, each of which has five choices. Each question has exactly one correct answer.

William guesses randomly at each answer. What is the probability that he gets six or fewer questions correct?

To find the probability that William gets six or fewer questions correct, we need to find the probability of getting each possible number of correct answers (0, 1, 2, 3, 4, 5, or 6) and then sum them up.

The probability of getting exactly k correct answers out of 8 can be calculated using the binomial probability formula:

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

Where:
- P(X = k) is the probability of getting k correct answers
- C(n, k) is the number of ways to choose k items from a set of n items (binomial coefficient)
- p is the probability of getting a single question correct
- n is the total number of questions

In this case, the probability of getting a single question correct (p) is 1 out of 5, since there are 5 choices for each question. So p = 1/5.

Let's calculate the probabilities for each possible number of correct answers and sum them up:

P(X ≤ 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)

P(X ≤ 6) = [C(8, 0) * (1/5)^0 * (4/5)^8] + [C(8, 1) * (1/5)^1 * (4/5)^7] + [C(8, 2) * (1/5)^2 * (4/5)^6] + [C(8, 3) * (1/5)^3 * (4/5)^5] + [C(8, 4) * (1/5)^4 * (4/5)^4] + [C(8, 5) * (1/5)^5 * (4/5)^3] + [C(8, 6) * (1/5)^6 * (4/5)^2]

Now we can calculate each term individually:

P(X ≤ 6) = [1 * (1/5)^0 * (4/5)^8] + [8 * (1/5)^1 * (4/5)^7] + [28 * (1/5)^2 * (4/5)^6] + [56 * (1/5)^3 * (4/5)^5] + [70 * (1/5)^4 * (4/5)^4] + [56 * (1/5)^5 * (4/5)^3] + [28 * (1/5)^6 * (4/5)^2]

Now, plug in the values and simplify each term:

P(X ≤ 6) = (1 * 1 * 0.32768) + (8 * 0.2 * 0.131072) + (28 * 0.04 * 0.0512) + (56 * 0.008 * 0.2048) + (70 * 0.0016 * 0.4096) + (56 * 0.00032 * 0.512) + (28 * 0.000064 * 0.64)

P(X ≤ 6) = 0.32768 + 0.2097152 + 0.0118784 + 0.0928256 + 0.045056 + 0.009216 + 0.001792

P(X ≤ 6) ≈ 0.6981

Therefore, the probability that William gets six or fewer questions correct is approximately 0.6981 or 69.81%.

To find the probability that William gets six or fewer questions correct, we need to find the probability of each possible outcome and then add them up.

Let's break it down step-by-step:

Step 1: Probability of getting one question correct:
Since each question has five choices, the probability of guessing the correct answer for one question is 1/5.

Step 2: Probability of getting one question wrong:
The probability of guessing the incorrect answer for one question is 4/5.

Step 3: Probability of getting six questions correct:
To get exactly six questions correct, William needs to guess correctly for 6 questions and incorrectly for the remaining 2 questions.
The probability of getting 6 questions correct is (1/5)^6 * (4/5)^2.

Step 4: Probability of getting five questions correct:
To get exactly five questions correct, William needs to guess correctly for 5 questions and incorrectly for the remaining 3 questions.
The probability of getting 5 questions correct is (1/5)^5 * (4/5)^3.

Step 5: Probability of getting four questions correct:
To get exactly four questions correct, William needs to guess correctly for 4 questions and incorrectly for the remaining 4 questions.
The probability of getting 4 questions correct is (1/5)^4 * (4/5)^4.

Step 6: Probability of getting three questions correct:
To get exactly three questions correct, William needs to guess correctly for 3 questions and incorrectly for the remaining 5 questions.
The probability of getting 3 questions correct is (1/5)^3 * (4/5)^5.

Step 7: Probability of getting two questions correct:
To get exactly two questions correct, William needs to guess correctly for 2 questions and incorrectly for the remaining 6 questions.
The probability of getting 2 questions correct is (1/5)^2 * (4/5)^6.

Step 8: Probability of getting one question correct:
To get exactly one question correct, William needs to guess correctly for 1 question and incorrectly for the remaining 7 questions.
The probability of getting 1 question correct is (1/5)^1 * (4/5)^7.

Step 9: Probability of getting zero questions correct:
To get exactly zero questions correct, William needs to guess incorrectly for all 8 questions.
The probability of getting 0 questions correct is (4/5)^8.

Step 10: Add up the probabilities:
To find the probability of getting six or fewer questions correct, we add up the probabilities from steps 3 to 9:
Probability = (1/5)^6 * (4/5)^2 + (1/5)^5 * (4/5)^3 + (1/5)^4 * (4/5)^4 + (1/5)^3 * (4/5)^5 + (1/5)^2 * (4/5)^6 + (1/5)^1 * (4/5)^7 + (4/5)^8.

Now we can calculate the probability.