A student takes a 20 question, multiple-choice exam with five choices for each question and guesses on each question. Find the probability of guessing at least  17 out of  20 correctly. Round your answer to three decimal places.

This problem involves a binomial probability distribution, as we have a fixed number of trials (20 questions), two possible outcomes (correct or incorrect), and the probability of success is constant for each trial.

Here are the steps to solve this:

### Given:
- Number of questions (\( n \)) = 20
- Probability of guessing correctly (\( p \)) = \( \frac{1}{5} \) = 0.2
- Probability of guessing incorrectly (\( q \)) = 1 - 0.2 = 0.8
- We're interested in the probability of guessing at least 17 questions correctly, so we want \( P(X \geq 17) \), where \( X \) is the number of correct answers.

The binomial probability formula is:

\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot q^{n-k}
\]

Where:
- \( \binom{n}{k} \) is the binomial coefficient \( \frac{n!}{k!(n-k)!} \),
- \( p^k \) is the probability of guessing \( k \) questions correctly,
- \( q^{n-k} \) is the probability of guessing \( n-k \) questions incorrectly.

We are looking for \( P(X \geq 17) \), which is the sum of probabilities for 17, 18, 19, and 20 correct answers:

\[
P(X \geq 17) = P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20)
\]

I will now calculate these probabilities.The probability of guessing at least 17 out of 20 questions correctly by pure guessing is extremely small—so small, in fact, that it rounds to 0 when computed to three decimal places.

This shows that achieving such a result by chance is practically impossible.

Would you like further details, or do you have any other questions?

Here are five related questions for further exploration:
1. What is the probability of guessing exactly 16 questions correctly?
2. How does the probability change if there are only four choices per question?
3. What is the expected number of correct answers when guessing on a 20-question exam?
4. How would the probability change if the student knew the correct answer for half the questions?
5. Can we calculate the standard deviation for the number of correct guesses?

**Tip:** The binomial distribution is ideal for scenarios with fixed numbers of independent trials, each having the same probability of success.

A student takes a 20 question, multiple-choice exam with five choices for each question and guesses on each question. Find the probability of guessing at least 17 out of 20 correctly. Round your answer to three decimal places.

This detailed solution shows how to calculate the probability of guessing at least 17 correct answers on a 20-question multiple-choice exam with five choices per question. The problem is solved using the binomial distribution formula and is ideal for students in Grades 10-12.

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