Two urns respectively contain 3 white, 7 red, 15 black balls and 10 white, 6 red, 9 black balls. One ball is taken from each urn. Find the probability that both be of the same colour.

Answer

To solve the problem of finding the probability that both balls drawn from two urns are of the same color, we can follow these steps:

### Step 1: Identify the contents of each urn
- **Urn 1 (U1)** contains:
- 3 white balls
- 7 red balls
- 15 black balls
- **Urn 2 (U2)** contains:
- 10 white balls
- 6 red balls
- 9 black balls

### Step 2: Calculate the total number of balls in each urn
- For Urn 1:
\[ \text{Total in U1} = 3 + 7 + 15 = 25 \]
- For Urn 2:
\[ \text{Total in U2} = 10 + 6 + 9 = 25 \]

### Step 3: Determine the probability of drawing balls of the same color
We need to calculate the probabilities for each color and then sum them up.

1. **Probability of drawing white balls:**
\[ P(\text{White}) = \frac{3}{25} \times \frac{10}{25} = \frac{30}{625} \]

2. **Probability of drawing red balls:**
\[ P(\text{Red}) = \frac{7}{25} \times \frac{6}{25} = \frac{42}{625} \]

3. **Probability of drawing black balls:**
\[ P(\text{Black}) = \frac{15}{25} \times \frac{9}{25} = \frac{135}{625} \]

### Step 4: Sum the probabilities of the same color
Now, we add the probabilities of drawing balls of the same color:
\[ P(\text{Same Color}) = P(\text{White}) + P(\text{Red}) + P(\text{Black}) = \frac{30}{625} + \frac{42}{625} + \frac{135}{625} \]
\[ P(\text{Same Color}) = \frac{30 + 42 + 135}{625} = \frac{207}{625} \]

### Step 5: Final probability
Thus, the probability that both balls drawn from the urns are of the same color is:
\[ \frac{207}{625} \]