An urn contains 4 white 6 black and 8 red balls. If 3 balls are drawn one by one without replacement, find the probability of getting all white balls.

Options

A. `5/204`
B. `1/204`
C. `13//204`
D. none of these

Answer

To solve the problem of finding the probability of drawing 3 white balls from an urn containing 4 white, 6 black, and 8 red balls without replacement, we can follow these steps:

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

**Total number of balls = 4 + 6 + 8 = 18 balls.**

### Step 2: Calculate the probability of drawing the first white ball.
When we draw the first ball, there are 4 white balls out of a total of 18 balls.

**Probability of drawing the first white ball (P(A)) = Number of favorable outcomes / Total outcomes = 4 / 18 = 2 / 9.**

### Step 3: Calculate the probability of drawing the second white ball.
After drawing the first white ball, there will be:
- 3 white balls left
- 17 balls remaining in total (since one ball has been drawn).

**Probability of drawing the second white ball (P(B|A)) = Number of favorable outcomes / Total outcomes = 3 / 17.**

### Step 4: Calculate the probability of drawing the third white ball.
After drawing the second white ball, there will be:
- 2 white balls left
- 16 balls remaining in total.

**Probability of drawing the third white ball (P(C|A and B)) = Number of favorable outcomes / Total outcomes = 2 / 16 = 1 / 8.**

### Step 5: Calculate the combined probability of drawing three white balls.
The combined probability of drawing three white balls in succession is the product of the individual probabilities calculated in the previous steps.

**P(A ∩ B ∩ C) = P(A) * P(B|A) * P(C|A and B)**

Substituting the values we found:
\[ P(A ∩ B ∩ C) = \left(\frac{2}{9}\right) \times \left(\frac{3}{17}\right) \times \left(\frac{1}{8}\right) \]

### Step 6: Perform the multiplication.
Calculating this gives:
\[ P(A ∩ B ∩ C) = \frac{2 \times 3 \times 1}{9 \times 17 \times 8} = \frac{6}{1224} \]

### Step 7: Simplify the fraction.
Now, simplify \( \frac{6}{1224} \):
\[ \frac{6}{1224} = \frac{1}{204} \]

### Final Answer:
The probability of drawing all three white balls is \( \frac{1}{204} \).

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