
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A certain province began to fully implement a new college entrance examination plan in 2021. In the six elective subjects, Physics and History use the original score for scoring; while Political Science, Geography, Chemistry, and Biology use grade conversion for scoring. The original scores of each student in these four subjects are divided into five levels from high to low: A, B, C, D, E, with the proportions of students in each level being 15%, 35%, 35%, 13%, and 2% respectively. The scores are then converted according to a given formula. The province organized a unified exam for the first year students and converted the original scores of Political Science, Geography, Chemistry, and Biology into grades.

1. In the Biology subject at a certain school, there are a total of 10 students who received an A grade. Their original scores and converted scores are shown in the table below:

| Original Score | 91 | 90 | 89 | 88 | 87 | 85 | 83 | 82 |
|----------------|----|----|----|----|----|----|----|----|
| Converted Score| 100| 99 | 97 | 95 | 94 | 91 | 88 | 86 |
| Number of Students | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 |

Now, from these 10 students, 3 students are randomly selected. Let X be the number of students among the 3 whose Biology converted score is not less than 95. Find the probability distribution and the expected value of X.

2. Assuming that the original scores Y of Biology for the first-year students in this province follow a normal distribution $N(75.8, 36)$. If $Y \sim N(\mu, \sigma^2)$, let $η = \frac{Y - \mu}{\sigma}$. Then $η \sim N(0, 1)$. Solve the following problems:

   i. If the minimum score for grade C in Biology is taken as the cutoff point for stratified teaching, estimate the cutoff point (rounded to the nearest integer).

   ii. Now, the original scores of Biology for 800 first-year students in the province are randomly selected. If these students' scores are independent, let ξ be the number of students with scores not less than 71. Find the value of k where $P(\xi = k)$ is maximized.

Given: $P(\eta \leq 0.8) \approx 0.788$, $P(\eta \leq 1.04) \approx 0.85$.

Present the answer in LaTex format: \boxed{Your answer}
user