Two balls are drawn from an urn containing 2 white , 3 red and 4 black balls , one by one without replacement . What is the probability that
both balls are of the same colour ,

Answer

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

### Step 1: Determine the total number of balls in the urn.
The urn contains:
- 2 white balls
- 3 red balls
- 4 black balls

Total number of balls = 2 (white) + 3 (red) + 4 (black) = 9 balls.

### Step 2: Calculate the probability of drawing two balls of the same color.
We need to consider three cases: both balls being white, both being red, and both being black.

#### Case 1: Both balls are white.
- Probability of drawing the first white ball = Number of white balls / Total number of balls = 2/9.
- After drawing one white ball, there is 1 white ball left and 8 balls total.
- Probability of drawing the second white ball = Number of remaining white balls / Total remaining balls = 1/8.

Thus, the probability of both balls being white:
\[ P(\text{White, White}) = \frac{2}{9} \times \frac{1}{8} = \frac{2}{72}. \]

#### Case 2: Both balls are red.
- Probability of drawing the first red ball = Number of red balls / Total number of balls = 3/9 = 1/3.
- After drawing one red ball, there are 2 red balls left and 8 balls total.
- Probability of drawing the second red ball = Number of remaining red balls / Total remaining balls = 2/8 = 1/4.

Thus, the probability of both balls being red:
\[ P(\text{Red, Red}) = \frac{3}{9} \times \frac{2}{8} = \frac{6}{72}. \]

#### Case 3: Both balls are black.
- Probability of drawing the first black ball = Number of black balls / Total number of balls = 4/9.
- After drawing one black ball, there are 3 black balls left and 8 balls total.
- Probability of drawing the second black ball = Number of remaining black balls / Total remaining balls = 3/8.

Thus, the probability of both balls being black:
\[ P(\text{Black, Black}) = \frac{4}{9} \times \frac{3}{8} = \frac{12}{72}. \]

### Step 3: Add the probabilities of the three cases.
Now, we add the probabilities of the three cases:
\[ P(\text{Same Color}) = P(\text{White, White}) + P(\text{Red, Red}) + P(\text{Black, Black}) = \frac{2}{72} + \frac{6}{72} + \frac{12}{72} = \frac{20}{72}. \]

### Step 4: Simplify the probability.
To simplify \(\frac{20}{72}\):
\[ \frac{20}{72} = \frac{5}{18} \quad \text{(dividing both numerator and denominator by 4)}. \]

### Final Answer:
The probability that both balls drawn are of the same color is \(\frac{5}{18}\).

---