An urn contains 4 white balls, 6 black balls and 8 red balls. Three balls are drawn one after another without replacement. What is the probability that all three balls drawn are white?

Introduction / Context:
This is a classic probability problem involving drawing balls from an urn without replacement. We are given the counts of white, black and red balls and asked to find the probability that all three balls drawn are white. Because the drawing is without replacement, the probabilities change after each draw, and we must either use combinations or sequential multiplication of conditional probabilities to find the answer.

Given Data / Assumptions:

White balls in the urn: 4.

Black balls: 6.

Red balls: 8.

Total balls = 4 + 6 + 8 = 18.

Three balls are drawn one after another without replacement.

We want the event that all three drawn balls are white.

Concept / Approach:
There are two standard approaches. One uses combinations to count the number of ways to choose three balls from the urn and then restrict to white balls. The other uses the multiplication rule for successive probabilities, taking into account that after each white ball is drawn, both the number of white balls and the total number of balls decrease. Both approaches should lead to the same probability.

Step-by-Step Solution (Combinational Method):
Step 1: Total number of ways to choose any 3 balls from 18 = 18C3.Step 2: Compute 18C3 = (18 * 17 * 16) / (3 * 2 * 1) = 4896 / 6 = 816.Step 3: Number of favourable ways: choose 3 balls from the 4 white balls = 4C3.Step 4: Compute 4C3 = 4.Step 5: Probability = favourable / total = 4 / 816.Step 6: Simplify 4 / 816 = 1 / 204.

Step-by-Step Solution (Sequential Probability):
Step 1: Probability first ball is white = 4 / 18.Step 2: After drawing one white ball, white balls remaining = 3 and total balls = 17. Probability second ball is white = 3 / 17.Step 3: After drawing two white balls, white balls remaining = 2 and total balls = 16. Probability third ball is white = 2 / 16.Step 4: Multiply: overall probability = (4 / 18) * (3 / 17) * (2 / 16) = (4 * 3 * 2) / (18 * 17 * 16).Step 5: Simplify numerator and denominator to confirm that this also equals 1 / 204.

Verification / Alternative check:
Reducing (4 * 3 * 2) / (18 * 17 * 16) step by step gives 24 / (4896). Dividing numerator and denominator by 24, we get 1 / 204, matching the combinational method. This confirms that both approaches are consistent and that the probability is correctly simplified.

Why Other Options Are Wrong:
5/204 and 13/204 are larger numerators but do not correspond to any meaningful count of favourable combinations. The fractions 3/68 and 1/68 have different denominators and would imply a completely different distribution of balls. Only 1/204 aligns with both the combination formula and the stepwise probability calculation.

Common Pitfalls:
Students sometimes forget that the draws are without replacement and treat each draw as 4/18, leading to (4/18)^3, which is incorrect. Others miscalculate 18C3 or 4C3, producing an incorrect denominator or numerator. Carefully distinguishing between with and without replacement and double-checking combination calculations are essential to avoid these mistakes.

Final Answer:
The probability that all three drawn balls are white is 1/204.