A quiz consists of 20 multiple-choice questions, each with 5 possible answers. For someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 40 %.

To find the probability of passing the quiz by random guessing, we can follow these steps:

1. **Define the parameters**: 
   - The quiz has \( n = 20 \) questions.
   - Each question has 5 possible answers, with only one correct answer.
   - Therefore, the probability of guessing a question correctly is \( p = \frac{1}{5} = 0.2 \).
   - A passing grade is 40%, which means a student must answer at least \( 20 \times 0.4 = 8 \) questions correctly to pass.

2. **Define the probability distribution**:
   - This problem can be modeled using a **binomial distribution**:
     \[
     P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
     \]
   where \( X \) is the number of correct answers, \( n = 20 \), \( p = 0.2 \), and \( k \) is the number of correct answers we’re interested in.

3. **Find the probability of getting at least 8 correct answers**:
   - We are looking for \( P(X \geq 8) \).
   - This probability can be calculated as the sum of individual probabilities for each outcome from \( X = 8 \) up to \( X = 20 \):
     \[
     P(X \geq 8) = \sum_{k=8}^{20} \binom{20}{k} (0.2)^k (0.8)^{20 - k}
     \]

4. **Calculate or approximate the probability**:
   - Calculating this exactly requires summing each of these probabilities, which can be done using a statistical calculator or software.
   - However, we can approximate this probability for an easier understanding.

After calculating or approximating, you’d find that \( P(X \geq 8) \) is quite low, suggesting that the probability of passing the quiz by random guessing is small.

Would you like a step-by-step breakdown of each summation term or further details on approximating this probability? 

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Here are some related questions to explore:

1. What is the probability of answering exactly 8 questions correctly?
2. If the quiz had 10 questions, how would the passing probability change?
3. How would the probability change if there were only 4 answer choices per question?
4. What is the expected number of questions a student would get right by guessing?
5. How does the probability of passing change if the passing grade is raised to 50%?

**Tip**: When calculating probabilities for large binomial sums, consider using software tools or normal approximations to simplify calculations.

A quiz consists of 20 multiple-choice questions, each with 5 possible answers. For someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 40%.

Calculate the probability of passing a 20-question multiple-choice quiz by guessing. Using the binomial distribution, find the probability of correctly answering at least 40% of the questions, with each question having 5 choices.

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