To solve this, let's define the probability of William guessing each question correctly and then use the binomial probability formula to find the probability that he gets four or fewer questions correct.

Step 1: Define the Probability of Success and Failure

1. Probability of guessing a question correctly, $p$, is $\frac{1}{4} = 0.25$.
2. Probability of guessing a question incorrectly, $q$, is $1 - p = 0.75$.

Step 2: Define the Random Variable and Its Distribution

Let $X$ be the number of questions William gets correct. $X$ follows a binomial distribution with parameters:

• Number of trials $n = 6$ (since there are six questions),
• Probability of success $p = 0.25$.

Thus, $X \sim \text{Binomial}(n=6, p=0.25)$.

Step 3: Calculate the Probability $P(X \leq 4)$

We need to find the probability that William gets four or fewer questions correct, which can be written as: $P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)$

Using the binomial probability formula: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$

we calculate each probability from $P(X = 0)$ to $P(X = 4)$:

1. $P(X = 0) = \binom{6}{0} (0.25)^0 (0.75)^6$
2. $P(X = 1) = \binom{6}{1} (0.25)^1 (0.75)^5$
3. $P(X = 2) = \binom{6}{2} (0.25)^2 (0.75)^4$
4. $P(X = 3) = \binom{6}{3} (0.25)^3 (0.75)^3$
5. $P(X = 4) = \binom{6}{4} (0.25)^4 (0.75)^2$

Let’s calculate each term and sum them up.

Calculating:

1. $P(X = 0) \approx 0.1779785$
2. $P(X = 1) \approx 0.355957$
3. $P(X = 2) \approx 0.2966309$
4. $P(X = 3) \approx 0.131836$
5. $P(X = 4) \approx 0.032959$

Adding these together: $P(X \leq 4) \approx 0.1779785 + 0.355957 + 0.2966309 + 0.131836 + 0.032959 = 0.9954$

Answer

The probability that William gets four or fewer questions correct is approximately 0.9954.

Would you like a breakdown of each step or further details?

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1. How is binomial distribution applied in real-world situations?
2. What is the difference between cumulative and single-event probabilities?
3. How can you calculate similar probabilities for different values of $n$ and $p$?
4. What does the probability result imply in terms of expectation?
5. How does changing the number of choices per question affect the probability?

Tip: Always check if a problem requires cumulative probability; it’s essential for "at most" or "at least" types of questions.