Question

What is the probability of drawing two marbles of the same color from a bag containing 9 red, 9 blue, and 9 yellow marbles?

Answer 1

To find the probability of drawing two marbles of the same color from a bag containing 9 red, 9 blue, and 9 yellow marbles, we first need to determine the total number of marbles and the number of ways to draw two marbles of the same color.

### Total Number of Marbles
The bag contains:
- 9 red marbles
- 9 blue marbles
- 9 yellow marbles

Total number of marbles is:
\[
9 + 9 + 9 = 27
\]

### Total Ways to Draw Two Marbles
The number of ways to choose 2 marbles from 27 can be calculated using the combination formula:
\[
\binom{n}{r} = \frac{n!}{r!(n - r)!}
\]
So, the number of ways to choose 2 marbles from 27 is:
\[
\binom{27}{2} = \frac{27 \times 26}{2 \times 1} = \frac{702}{2} = 351
\]

### Ways to Draw Two Marbles of the Same Color
Next, we need to find the number of ways to draw 2 marbles of the same color. We can calculate this for each color:

1. **Red Marbles**:
\[
\binom{9}{2} = \frac{9 \times 8}{2} = \frac{72}{2} = 36
\]

2. **Blue Marbles**:
\[
\binom{9}{2} = \frac{9 \times 8}{2} = \frac{72}{2} = 36
\]

3. **Yellow Marbles**:
\[
\binom{9}{2} = \frac{9 \times 8}{2} = \frac{72}{2} = 36
\]

### Total Ways to Draw Two Marbles of the Same Color
Now, we add the ways to draw 2 marbles from each color:
\[
36 \text{ (red)} + 36 \text{ (blue)} + 36 \text{ (yellow)} = 108
\]

### Probability of Drawing Two Marbles of the Same Color
Finally, the probability can be calculated as the number of favorable outcomes (drawing 2 of the same color) divided by the total number of outcomes (drawing any 2 marbles):
\[
P(\text{same color}) = \frac{\text{Number of ways to draw 2 of the same color}}{\text{Total ways to draw 2 marbles}} = \frac{108}{351}
\]

To simplify this fraction:
\[
\frac{108 \div 9}{351 \div 9} = \frac{12}{39}
\]
\[
\frac{12 \div 3}{39 \div 3} = \frac{4}{13}
\]

Thus, the probability of drawing two marbles of the same color is:
\[
\boxed{\frac{4}{13}}
\]