There are `5` independent trials , probability of exactly one success is `0.4096` , probability of exactly `2` success is `0.2048`. Find probability of exactly `3 ` success

Answer

To solve the problem, we will use the binomial probability formula. The scenario describes independent trials with two possible outcomes: success and failure.

### Step-by-step Solution:

1. **Understanding the Binomial Probability Formula**:
The binomial probability formula for exactly \( r \) successes in \( n \) trials is given by:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]
where:
- \( n \) = number of trials
- \( r \) = number of successes
- \( p \) = probability of success
- \( q = 1 - p \) = probability of failure

2. **Setting Up the Problem**:
We have \( n = 5 \) trials. We are given:
- Probability of exactly 1 success:
\[ P(X = 1) = \binom{5}{1} p^1 q^4 = 0.4096 \]
- Probability of exactly 2 successes:
\[ P(X = 2) = \binom{5}{2} p^2 q^3 = 0.2048 \]

3. **Calculating the First Equation**:
Using \( P(X = 1) \):
\[ 5p q^4 = 0.4096 \quad \text{(Equation 1)} \]

4. **Calculating the Second Equation**:
Using \( P(X = 2) \):
\[ 10p^2 q^3 = 0.2048 \quad \text{(Equation 2)} \]

5. **Dividing Equation 1 by Equation 2**:
\[ \frac{5pq^4}{10p^2q^3} = \frac{0.4096}{0.2048} \]
Simplifying this gives:
\[ \frac{q}{2p} = 2 \quad \Rightarrow \quad q = 4p \]

6. **Using the fact that \( p + q = 1 \)**:
Since \( q = 4p \):
\[ p + 4p = 1 \quad \Rightarrow \quad 5p = 1 \quad \Rightarrow \quad p = \frac{1}{5} \]
Then,
\[ q = 1 - p = 1 - \frac{1}{5} = \frac{4}{5} \]

7. **Finding the Probability of Exactly 3 Successes**:
Now, we can find \( P(X = 3) \):
\[ P(X = 3) = \binom{5}{3} p^3 q^2 \]
Calculating this:
\[ P(X = 3) = 10 \left(\frac{1}{5}\right)^3 \left(\frac{4}{5}\right)^2 \]
\[ = 10 \cdot \frac{1}{125} \cdot \frac{16}{25} \]
\[ = 10 \cdot \frac{16}{3125} = \frac{160}{3125} = \frac{32}{625} \]

### Final Answer:
The probability of exactly 3 successes is:
\[ \frac{32}{625} \]