A multiple-choice test consists of 8 questions with 3 answers to each question (of which only one is correct). A student answers each question by rolling a balanced die and checking the first answer if he gets 1 or 2, the second answer if he gets 3 or 4 and the third answer if he gets 5 or 6. To get a distinction, the student must secure at least 75% correct answers. If there is no negative marking, what is the probability that the student secures a distinction?

b.

In a book of 520 pages, 390 typo-graphical errors occur. Considered Poisson law for the number of errors per page, find tile probability that a random sample of 5 pages will contain no error.

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### Part (a): Probability of Securing a Distinction

Let’s break down the probability of the student getting a distinction.

1. **Total Questions:** There are 8 questions.
2. **Answers per Question:** Each question has 3 possible answers, with only one being correct.
3. **Student’s Selection Method:** The student rolls a die to pick an answer, and each answer has a \( \frac{1}{3} \) probability of being selected. Thus, each answer the student picks has a probability of \( \frac{1}{3} \) of being correct.

4. **Defining a Distinction:** To achieve a distinction, the student must get at least 75% of the answers correct. This means the student needs to answer at least:
   \[
   8 \times 0.75 = 6 \text{ questions correctly.}
   \]

5. **Defining Success Probability for Each Question:** Each answer has a \( \frac{1}{3} \) probability of being correct.

6. **Modeling as a Binomial Distribution:** Let \( X \) be the number of correct answers. Then \( X \) follows a binomial distribution with parameters:
   \[
   X \sim \text{Binomial}(n = 8, p = \frac{1}{3})
   \]
   We want \( P(X \geq 6) \).

7. **Calculating \( P(X \geq 6) \):** 
   The probability of securing a distinction is:
   \[
   P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8)
   \]
   We can compute each term using the binomial probability formula:
   \[
   P(X = k) = \binom{8}{k} \left( \frac{1}{3} \right)^k \left( \frac{2}{3} \right)^{8-k}
   \]
   
   Calculating these values can give us the probability of achieving a distinction.

### Part (b): Probability of No Errors in 5 Pages

Let’s solve this using the Poisson distribution.

1. **Total Pages and Errors:** In a 520-page book, there are 390 typographical errors.
   
2. **Average Errors per Page (λ):** The average number of errors per page can be found by dividing the total errors by the total pages:
   \[
   \lambda = \frac{390}{520} = 0.75
   \]

3. **Using Poisson Distribution:** Let \( Y \) be the number of errors in a random sample of 5 pages. The Poisson distribution is suitable here, with a mean number of errors over 5 pages being:
   \[
   \lambda_{\text{5 pages}} = 5 \times 0.75 = 3.75
   \]
   Thus, \( Y \sim \text{Poisson}(3.75) \).

4. **Finding Probability of No Errors in 5 Pages:** We need \( P(Y = 0) \):
   \[
   P(Y = 0) = \frac{e^{-\lambda_{\text{5 pages}}} \lambda_{\text{5 pages}}^0}{0!} = e^{-3.75}
   \]
   Calculating \( e^{-3.75} \) will give us the probability that a random sample of 5 pages contains no errors.

Would you like to see these calculations in detail, or do you have any questions?

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### Additional Questions to Explore
1. How would the probability change in part (a) if there were 4 answers instead of 3?
2. What is the probability of getting exactly 5 correct answers in part (a)?
3. How would the outcome change if the student needed only a 50% score to pass?
4. How does the mean number of errors per page affect the Poisson probability in part (b)?
5. If we consider a larger sample of pages in part (b), how would the probability of no errors change?

### Tip
For Poisson problems, keep in mind that the mean number of events within the given interval is key for calculating probabilities directly using the Poisson formula.

A multiple-choice test consists of 8 questions with 3 answers to each question (of which only one is correct). A student answers each question by rolling a balanced die and checking the first answer if he gets 1 or 2, the second answer if he gets 3 or 4 and the third answer if he gets 5 or 6. To get a distinction, the student must secure at least 75% correct answers. If there is no negative marking, what is the probability that the student secures a distinction? In a book of 520 pages, 390 typographical errors occur. Considering Poisson law for the number of errors per page, find the probability that a random sample of 5 pages will contain no error.

Explore the probability of achieving a distinction on an 8-question multiple-choice test by modeling answers with a binomial distribution. Also, calculate the probability of no typographical errors in a 5-page sample using Poisson distribution. This problem involves key probability concepts and is suitable for students in Grades 11-12.

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