A6. From a group of twenty Environment $$ \operatorname{Studies}(E S) $$ students, ten are randomly selected for attachment. Let $$ X $$ denotes the number best $$ E S $$ students among the ten selected. (a) What is the probability that the ten selected include all of the fair five best $$ E S $$ students in a class of twenty. (b) Calculate $$ E(X) $$ and $$ \operatorname{Var}(X) $$.

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(a) The total number of ways of choosing 10 students from 20 is
$$
\binom{20}{10}.
$$
To have all five best $$E S$$ students included, we must choose these 5 and then choose the remaining 5 from the other 15 students. The number of favourable ways is
$$
\binom{15}{5}.
$$
Thus, the required probability is
$$
\frac{\binom{15}{5}}{\binom{20}{10}}.
$$

(b) The random variable $$X$$ (number of best $$E S$$ students selected) follows a hypergeometric distribution with parameters: $$N=20$$ (total students), $$K=5$$ (best students), and $$n=10$$ (students selected).

The expected value is
$$
E(X) = n\frac{K}{N} = 10\frac{5}{20} = 2.5.
$$

The variance is given by the formula
$$
\operatorname{Var}(X) = n \frac{K}{N} \left(1-\frac{K}{N}\right) \frac{N-n}{N-1}.
$$
Substitute the values:
$$
\operatorname{Var}(X) = 10 \cdot \frac{5}{20} \cdot \left(1-\frac{5}{20}\right) \cdot \frac{20-10}{20-1}.
$$
Simplify step by step:
$$
\frac{5}{20} = \frac{1}{4}, \quad 1-\frac{1}{4} = \frac{3}{4},
$$
$$
\operatorname{Var}(X) = 10 \cdot \frac{1}{4} \cdot \frac{3}{4} \cdot \frac{10}{19} = 10 \cdot \frac{3}{16} \cdot \frac{10}{19}.
$$
$$
\operatorname{Var}(X) = \frac{300}{16 \cdot 19} = \frac{300}{304} = \frac{75}{76}.
$$

Thus, the answers are:

(a) $$\displaystyle \frac{\binom{15}{5}}{\binom{20}{10}}$$.

(b) $$E(X)=2.5$$ and $$\operatorname{Var}(X)=\frac{75}{76}$$.
(a) The probability that all five best $$E S$$ students are selected is $$\frac{\binom{15}{5}}{\binom{20}{10}}$$. (b) The expected number of best $$E S$$ students selected is 2.5, and the variance is $$\frac{75}{76}$$.

A6. From a group of twenty Environment students, ten are randomly selected
for attachment. Let denotes the number best students among the ten selected.
(a) What is the probability that the ten selected include all of the fair five best
students in a class of twenty.
(b) Calculate and .

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A6. From a group of twenty Environment students, ten are randomly selected for attachment. Let denotes the number best students among the ten selected. (a) What is the probability that the ten selected include all of the fair five best students in a class of twenty. (b) Calculate and .