If two balls are drawn from a bag containing three red balls and four blue balls, find the probability that
(a) They are of the same colour
(b) They are of different colours

Answer

To solve the problem of finding the probability of drawing two balls from a bag containing three red balls and four blue balls, we will follow these steps:

### Step 1: Determine the Total Number of Balls
The total number of balls in the bag is:
- Red balls = 3
- Blue balls = 4
- Total = 3 + 4 = 7 balls

### Step 2: Calculate the Total Number of Ways to Draw 2 Balls
We can use the combination formula \( nCr \) to find the total number of ways to choose 2 balls from 7:
\[ \text{Total ways} = \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \]

### Step 3: Calculate the Probability of Drawing Two Balls of the Same Color
To find the probability of drawing two balls of the same color, we consider two cases:
1. Both balls are red.
2. Both balls are blue.

**Case 1: Both Balls are Red**
The number of ways to choose 2 red balls from 3:
\[ \text{Ways to choose 2 red balls} = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3 \]

**Case 2: Both Balls are Blue**
The number of ways to choose 2 blue balls from 4:
\[ \text{Ways to choose 2 blue balls} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

**Total Ways to Choose Balls of the Same Color**
\[ \text{Total same color ways} = 3 + 6 = 9 \]

**Probability of Same Color**
\[ P(\text{Same Color}) = \frac{\text{Total same color ways}}{\text{Total ways}} = \frac{9}{21} = \frac{3}{7} \]

### Step 4: Calculate the Probability of Drawing Two Balls of Different Colors
To find the probability of drawing two balls of different colors, we can consider the only possible combination:
- One red ball and one blue ball.

**Ways to Choose One Red and One Blue Ball**
\[ \text{Ways to choose 1 red and 1 blue} = \binom{3}{1} \times \binom{4}{1} = 3 \times 4 = 12 \]

**Probability of Different Colors**
\[ P(\text{Different Colors}) = \frac{\text{Ways to choose 1 red and 1 blue}}{\text{Total ways}} = \frac{12}{21} = \frac{4}{7} \]

### Final Answers
(a) The probability that both balls are of the same color is \( \frac{3}{7} \).

(b) The probability that the balls are of different colors is \( \frac{4}{7} \).

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