Three balls are drawn at random. There are 3 red balls and 7 blue balls. What is probability that exactly one red ball is chosen ?

To find the probability that exactly one red ball is chosen when three balls are drawn at random from a set of 3 red balls and 7 blue balls, we can follow these steps:

### Step 1: Total number of balls
There are a total of \( 3 \) red balls and \( 7 \) blue balls, so the total number of balls is:
\[
3 + 7 = 10 \text{ balls}
\]

### Step 2: Total number of ways to choose 3 balls
The total number of ways to choose 3 balls out of 10 is given by the combination formula \( \binom{n}{k} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose:
\[
\text{Total ways} = \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]

### Step 3: Number of favorable outcomes (exactly one red ball)
To get exactly one red ball, we need to choose:
- 1 red ball out of 3 red balls
- 2 blue balls out of 7 blue balls

The number of ways to choose 1 red ball from 3 is:
\[
\binom{3}{1} = 3
\]
The number of ways to choose 2 blue balls from 7 is:
\[
\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21
\]

Therefore, the number of favorable outcomes is:
\[
3 \times 21 = 63
\]

### Step 4: Probability calculation
The probability of drawing exactly one red ball is the ratio of the number of favorable outcomes to the total number of outcomes:
\[
\text{Probability} = \frac{63}{120} = \frac{21}{40}
\]

So, the probability that exactly one red ball is chosen is \( \frac{21}{40} \).

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?
2. How would the probability change if there were 5 red balls and 5 blue balls?
3. What is the probability of drawing no red balls at all?
4. Can you calculate the probability of drawing exactly three blue balls?
5. How does the probability change if we draw 4 balls instead of 3?

**Tip:** Remember that combination formulas \( \binom{n}{k} \) are very useful when dealing with problems involving selection without regard to the order.

Learn how to calculate the probability of drawing exactly one red ball when three balls are drawn at random from a set containing 3 red balls and 7 blue balls. Detailed step-by-step solution using combinatorics and probability theory. Suitable for Grades 9-12.

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