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To solve this problem, we need to analyze the scenario using probability theory, specifically binomial probability, because we are dealing with a series of independent trials (each question) where there are two possible outcomes (correct or incorrect answer). The situation given involves a multiple-choice test with 14 questions, each having 5 possible answers, so the probability of guessing a correct answer is \( p = \frac{1}{5} = 0.2 \).

### Step 1: Binomial Distribution Formula
The binomial probability of getting exactly \( k \) correct answers out of \( n \) questions is given by:

\[
P(k \text{ correct}) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k}
\]

Where:
- \( n = 14 \) (number of questions),
- \( p = 0.2 \) (probability of a correct answer),
- \( k \) is the number of correct answers,
- \( \binom{n}{k} \) is the binomial coefficient, or "n choose k."

### Step 2: Unusual Event
An event is typically considered unusual if the probability is less than 5%. Therefore, we need to find the smallest number of correct answers such that the probability of getting at least that many correct answers is less than 5%.

### Step 3: Calculate Cumulative Probabilities
We need to calculate the cumulative probabilities for different values of \( k \) (correct answers). We will calculate these for values starting at the highest (like 14) and move lower until we find a value where the cumulative probability exceeds 95%, meaning anything above that would be less than 5%.

Based on statistical tools or binomial distribution tables, this process will show that getting **6 or more** correct answers would be considered unusual.

Thus, the answer is **6**.

Would you like further details or clarifications on the solution? Here are some additional questions to expand on the topic:

1. What is the exact probability of getting 6 correct answers?
2. How does the binomial distribution work with varying numbers of questions?
3. Can you explain why getting 14 correct answers by guessing is even more unusual?
4. How do you calculate the cumulative probability for this distribution?
5. What would change if the test had more than 14 questions?

**Tip:** In probability theory, "unusual" often refers to outcomes in the tails of the distribution, which helps define rare or extreme events.

When taking a 14 question multiple choice test, where each question has 5 possible answers, it would be unusual to get ____ or more questions correct by guessing alone. Note: Of course it would be unusual to get 14 questions correct. You need to enter the smallest possible number of questions to make getting 'that many correct' unusual.

Explore how to determine the smallest number of correct answers considered unusual in a 14-question multiple-choice test. This problem involves binomial probability concepts and cumulative probabilities, suitable for students in Grades 10-12.

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