https://file.aiquickdraw.com/mathbot/file/1726010893366h3ptxskr.jpg

Let's work through the questions in the image.

### Given:
- 7 boys and 10 girls take an examination.
- Each student has a probability of 0.2 of failing the exam.

### Part (a) - **What is the sample space for this experiment?**

The sample space includes all possible outcomes for whether each student (17 total: 7 boys and 10 girls) passes or fails the examination. Each student can either pass or fail. Therefore, the size of the sample space is \( 2^{17} \), as each student has two possible outcomes (pass or fail). Thus, the sample space for this experiment is all combinations of the 17 students either passing or failing.

### Part (b) - **Given that three students failed the examination, what is the probability that all three are boys?**

We need to calculate the conditional probability that all three students who failed are boys.

- Total number of boys = 7
- Total number of girls = 10
- Total number of students = 17

We are selecting 3 students who failed, and we want all 3 to be boys.

To find the probability:
1. The total number of ways to choose 3 students who failed from 17 students is given by the combination:
   \[
   \binom{17}{3} = \frac{17 \times 16 \times 15}{3 \times 2 \times 1} = 680
   \]
2. The number of ways to choose 3 boys who failed from the 7 boys is:
   \[
   \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
   \]
3. The probability that all 3 who failed are boys is:
   \[
   P(\text{all 3 are boys}) = \frac{\binom{7}{3}}{\binom{17}{3}} = \frac{35}{680} = \frac{1}{19.43} \approx 0.0517
   \]

So, the probability that all three who failed are boys is approximately 0.0517, or about 5.17%.

### Part (c) - **What is the name of the probability distribution you are using?**

The distribution being used here is the **Hypergeometric distribution**. This distribution applies when you are selecting objects (in this case, students) without replacement from a finite population, and you want to know the probability of a certain number of successes (in this case, boys failing the exam) from a given number of trials (students failing).

### Part (d) - **If the probability of each failure is 0.8 instead of 0.2, what is the answer to

Seven boys and 10 girls take an examination, and each student has a probability of 0.2 of failing the examination.

(a) What is the sample space for this experiment?
(b) Given that three students failed the examination, what is the probability that all three are boys?
(c) What is the name of the probability distribution you are using?
(d) If the probability of each failure is 0.8 instead of 0.2, what is the answer to part b?

Explore a probability problem involving 7 boys and 10 girls taking an examination, where each has a 0.2 probability of failing. Learn how to determine the sample space, calculate the probability that all who failed are boys using the Hypergeometric Distribution, and understand the impact of changing the probability of failure.

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