To solve this problem, we need to calculate the probability of getting at least one question wrong on a 9-question multiple-choice test where each question has 4 possible answers.

Step-by-Step Solution

1. Probability of Getting One Question Right: Each question has 4 choices, and only 1 is correct. So, the probability of selecting the correct answer for any given question is: $\frac{1}{4}$

2. Probability of Getting All Questions Right: Since there are 9 questions and each question must be answered correctly to get them all right, the probability of getting all 9 questions correct is: $\left(\frac{1}{4}\right)^9 = \frac{1}{4^9}$

3. Probability of Getting At Least One Question Wrong: The probability of getting at least one question wrong is the complement of getting all questions right. So, we calculate this as: $1 - \frac{1}{4^9}$

4. Calculating $4^9$: $4^9 = 262144$, so: $1 - \frac{1}{262144} = \frac{262144 - 1}{262144} = \frac{262143}{262144}$

Final Answer

The probability of getting at least one question wrong is: $\frac{262143}{262144}$

Would you like further details on this solution or have any questions?

Here are some related questions for further exploration:

1. What would the probability be of getting exactly one question wrong on this test?
2. How would the probability change if each question had only 3 answers instead of 4?
3. What is the probability of getting at least two questions wrong on a similar test?
4. How does the probability change if the test has more questions, say 12 instead of 9?
5. How can we generalize this problem for a test with $n$ questions and $m$ answer choices per question?

Tip: In probability problems, using complements (like calculating the probability of "at least one wrong" by finding "all correct" first) often simplifies the calculations.