To solve this problem, we need to calculate the probability that a student gets more than 2 questions correct out of 8 when guessing randomly on each question.

Step 1: Determine the probability of guessing a question correctly.

Since there are 4 choices (a, b, c, d) and only one is correct, the probability of guessing a question correctly is: $p = \frac{1}{4} = 0.25$ The probability of guessing incorrectly is therefore: $q = 1 - p = 1 - 0.25 = 0.75$

Step 2: Use the binomial probability formula.

The probability of getting exactly $k$ questions correct out of $n$ total questions follows the binomial distribution: $P(X = k) = \binom{n}{k} p^k q^{n-k}$ where $\binom{n}{k}$ is the binomial coefficient.

Step 3: Calculate the probability of getting more than 2 questions correct.

We need to find $P(X > 2)$, which is: $P(X > 2) = 1 - P(X \leq 2)$ This requires calculating $P(X = 0)$, $P(X = 1)$, and $P(X = 2)$, and subtracting their sum from 1.

1. $P(X = 0)$: $P(X = 0) = \binom{8}{0} (0.25)^0 (0.75)^8 = 1 \times 1 \times 0.1001 = 0.1001$
2. $P(X = 1)$: $P(X = 1) = \binom{8}{1} (0.25)^1 (0.75)^7 = 8 \times 0.25 \times 0.1335 = 0.2670$
3. $P(X = 2)$: $P(X = 2) = \binom{8}{2} (0.25)^2 (0.75)^6 = 28 \times 0.0625 \times 0.1779785 = 0.3115$

Step 4: Add up the probabilities for $X \leq 2$ and subtract from 1.

$P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.1001 + 0.2670 + 0.3115 = 0.6786$ Thus, $P(X > 2) = 1 - 0.6786 = 0.3214$

Final Answer:

The probability that the student gets more than 2 questions correct is approximately 0.32 (rounded to two decimal places).

Would you like more details or have any questions?

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Related Questions:

1. What is the probability of getting exactly 3 questions correct?
2. How would the probability change if the test had 10 questions?
3. What is the expected number of correct answers if the student guesses on all questions?
4. How would the probability change if each question had 5 answer choices?
5. Can you calculate the probability of getting at least one question correct?
6. What is the variance of the number of correct answers?
7. How would the probability change if the student knew the correct answer to half of the questions?
8. How is the cumulative distribution function (CDF) related to this problem?

Tip: When dealing with binomial distributions, it's often helpful to check the cumulative distribution function (CDF) for quick probabilities of "at least" or "at most" scenarios.