There are 8 white and 7 black balls in first bag and 9 white and 6 black balls are in second bag. One ball is drawn from each bag. Find the probability that both are of the same colour.

Explanation

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

### Step 1: Determine the total number of balls in each bag.
- **First Bag**: 8 white balls + 7 black balls = 15 balls
- **Second Bag**: 9 white balls + 6 black balls = 15 balls

### Step 2: Calculate the probability of drawing two white balls.
- Probability of drawing a white ball from the first bag = Number of white balls in the first bag / Total number of balls in the first bag = 8/15
- Probability of drawing a white ball from the second bag = Number of white balls in the second bag / Total number of balls in the second bag = 9/15
- Since these events are independent, the combined probability of both events happening (drawing two white balls) is:
\[ P(\text{both white}) = P(\text{white from bag 1}) \times P(\text{white from bag 2}) = \frac{8}{15} \times \frac{9}{15} = \frac{72}{225} \]

### Step 3: Calculate the probability of drawing two black balls.
- Probability of drawing a black ball from the first bag = Number of black balls in the first bag / Total number of balls in the first bag = 7/15
- Probability of drawing a black ball from the second bag = Number of black balls in the second bag / Total number of balls in the second bag = 6/15
- The combined probability of both events happening (drawing two black balls) is:
\[ P(\text{both black}) = P(\text{black from bag 1}) \times P(\text{black from bag 2}) = \frac{7}{15} \times \frac{6}{15} = \frac{42}{225} \]

### Step 4: Add the probabilities of both events.
- Since the events of drawing two white balls and two black balls are mutually exclusive, we can add their probabilities:
\[ P(\text{same color}) = P(\text{both white}) + P(\text{both black}) = \frac{72}{225} + \frac{42}{225} = \frac{114}{225} \]

### Step 5: Simplify the final probability.
- To simplify \(\frac{114}{225}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 3:
\[ \frac{114 \div 3}{225 \div 3} = \frac{38}{75} \]

### Final Answer:
The probability that both balls drawn are of the same color is \(\frac{38}{75}\).

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