Toán học - Đại học

Let's solve each problem.

**Problem 8:**

We have a balanced die rolled 10 times. Let \(X\) be the number of times the die shows 3 dots.

This is a binomial distribution problem with:
- Number of trials \(n = 10\).
- Probability of success for each trial \(p = \frac{1}{6}\).

The probability distribution of \(X\) is:
\[
P(X = k) = \binom{10}{k} \left(\frac{1}{6}\right)^k \left(\frac{5}{6}\right)^{10-k}
\]
where \(k = 0, 1, 2, \ldots, 10\).

The expected value \(E(X) = np = 10 \times \frac{1}{6} = \frac{10}{6} = \frac{5}{3}\).

**Problem 9:**

We have 100 seeds, each seed germinates with probability 0.9. Let \(X\) be the number of seeds that germinate.

This is also a binomial distribution problem with:
- Number of trials \(n = 100\).
- Probability of success for each trial \(p = 0.9\).

The probability distribution of \(X\) is:
\[
P(X = k) = \binom{100}{k} (0.9)^k (0.1)^{100-k}
\]
where \(k = 0, 1, 2, \ldots, 100\).

The expected value \(E(X) = np = 100 \times 0.9 = 90\).

The variance \(\text{Var}(X) = np(1-p) = 100 \times 0.9 \times 0.1 = 9\).