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This problem involves a binomial distribution, where the probability of \( k \) successes in \( n \) trials is given by the binomial formula:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

Given:
- \( n = 5 \) (5 independent trials)
- Probabilities for 1 and 2 successes are:
  - \( P(X = 1) = 0.4096 \)
  - \( P(X = 2) = 0.2048 \)

We need to find the parameter \( p \), which is the probability of success in a single trial.

### Step 1: Formula for 1 success \( P(X = 1) \)
Using the binomial formula for \( X = 1 \):
\[
P(X = 1) = \binom{5}{1} p (1 - p)^4
\]
\[
0.4096 = 5p (1 - p)^4
\]

### Step 2: Formula for 2 successes \( P(X = 2) \)
Similarly, for \( X = 2 \):
\[
P(X = 2) = \binom{5}{2} p^2 (1 - p)^3
\]
\[
0.2048 = 10 p^2 (1 - p)^3
\]

### Step 3: Solve for \( p \)
Now we have two equations:
\[
5p (1 - p)^4 = 0.4096
\]
\[
10 p^2 (1 - p)^3 = 0.2048
\]

By dividing these two equations, we can eliminate some terms and solve for \( p \):

\[
\frac{5p (1 - p)^4}{10 p^2 (1 - p)^3} = \frac{0.4096}{0.2048}
\]
\[
\frac{(1 - p)}{2p} = 2
\]
Multiplying both sides by \( 2p \), we get:
\[
1 - p = 4p
\]
\[
1 = 5p
\]
\[
p = 0.2
\]

Thus, the parameter \( p \) is \( \boxed{0.20} \), corresponding to option \( \text{c} \).

---

Would you like more details or have any other questions?

Here are 5 related questions for further understanding:
1. How is the binomial distribution derived?
2. What is the role of the binomial coefficient in this context?
3. How would the probabilities change if the number of trials \( n \) increased?
4. Can the binomial formula be used for continuous data?
5. How would you calculate the expected value of this binomial distribution?

**Tip:** In binomial problems, always carefully match the formula with the given probability for a certain number of successes!

In a binomial distribution consisting of 5 independent trials, probabilities of 1 and 2 successes are 0.4096 and 0.2048 respectively. Find the parameter p of the distribution.

Solve a binomial distribution problem with 5 independent trials, given probabilities for 1 and 2 successes. This solution involves key probability concepts and the binomial distribution formula to find the parameter p. Suitable for students in Grades 10-12.

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