Three balls are drawn at random with replacement. There are 3 red balls and 7 blue balls. What is probability that exactly one red ball is chosen ? Give answer in decimal. 

When drawing with replacement, the outcome of each draw is independent, meaning the composition of the pool of balls remains the same after each draw. Here's how we can calculate the probability that exactly one red ball is chosen:

### Step 1: Probability of drawing a red or blue ball
- Probability of drawing a red ball \( P(\text{Red}) \) is:
\[
P(\text{Red}) = \frac{3}{10} = 0.3
\]
- Probability of drawing a blue ball \( P(\text{Blue}) \) is:
\[
P(\text{Blue}) = \frac{7}{10} = 0.7
\]

### Step 2: Probability of exactly one red ball in three draws
To have exactly one red ball, we must have one red ball and two blue balls. This can happen in three different ways:
- Red on the first draw, Blue on the second and third draw
- Blue on the first draw, Red on the second, Blue on the third draw
- Blue on the first and second draw, Red on the third draw

The probability for each of these scenarios is:
\[
P(\text{Red, Blue, Blue}) = P(\text{Red}) \times P(\text{Blue}) \times P(\text{Blue}) = 0.3 \times 0.7 \times 0.7 = 0.147
\]

Since there are three different sequences that can result in exactly one red ball, the total probability is:
\[
P(\text{Exactly 1 Red}) = 3 \times 0.147 = 0.441
\]

### Final Answer:
The probability that exactly one red ball is chosen is \( 0.441 \).

Would you like any details or have any questions? Here are five related questions that might interest you:

1. What is the probability of drawing exactly two red balls with replacement?
2. How would the probability change if there were 5 red balls and 5 blue balls?
3. What is the probability of drawing at least one red ball?
4. What is the probability of drawing three blue balls in a row?
5. How does the probability change if the draws are without replacement?

**Tip:** For problems with replacement, remember that the probabilities stay constant across each draw, making calculations simpler.

Understand the probability of drawing exactly one red ball when three balls are drawn at random with replacement. Detailed step-by-step solution and calculations provided.

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