A box contains 5 green pearls, 4 yellow pearls and 5 white pearls. Four pearls are drawn at random without replacement. What is the probability that the four pearls drawn are not all of the same colour?

Introduction / Context:
This probability question involves drawing multiple objects from a box containing several colours and asking for the probability that all selected objects are not of the same colour. It is more convenient to compute the complement event, namely that all four pearls are of the same colour, and subtract this from 1. This complement method is very useful when the direct event has many complicated cases.

Given Data / Assumptions:

• Green pearls: 5.
• Yellow pearls: 4.
• White pearls: 5.
• Total pearls: 5 + 4 + 5 = 14.
• Four pearls are drawn at random without replacement.
• We want the probability that the four pearls drawn are not all the same colour.

Concept / Approach:
Let us define event A as the event that all four pearls drawn are of the same colour. The required probability is then 1 minus P(A). We compute P(A) by summing probabilities of the disjoint subcases where all four are green, or all four are yellow, or all four are white. Each of these probabilities is found using combinations, because the order of drawing does not matter.

Step-by-Step Solution:
Step 1: Total ways to choose 4 pearls out of 14 is 14C4.Step 2: Compute 14C4 = (14 × 13 × 12 × 11) / (4 × 3 × 2 × 1) = 1001.Step 3: Ways to choose 4 green pearls: since there are 5 green pearls, this is 5C4 = 5.Step 4: Ways to choose 4 yellow pearls: there are 4 yellow pearls, so 4C4 = 1.Step 5: Ways to choose 4 white pearls: there are 5 white pearls, so again 5C4 = 5.Step 6: Total favourable ways for event A (all same colour) = 5 + 1 + 5 = 11.Step 7: Thus, P(A) = 11 / 1001 = 1 / 91.Step 8: Probability that the four pearls are not all of the same colour is 1 - 1 / 91 = 90 / 91.
Verification / Alternative check:
The computed probability 90 / 91 is very close to 1, which makes intuitive sense because it is quite rare to randomly pick four pearls all of the same colour given the distribution in the box. There are only 11 combinations out of 1001 where all four are of the same colour, confirming that the complement event is highly likely. This supports the correctness of the calculation.

Why Other Options Are Wrong:
The provided options in the original statement map as follows: Option A = 11/91, Option B = 4/11, Option C = 1/11 and Option D = 90/91. We have just shown that the correct probability is 90/91, so only Option D matches. The others represent either the complement probability or unrelated fractions and therefore are incorrect for the required event.

Common Pitfalls:
A frequent mistake is to misinterpret the phrase "not of the same colour" and attempt to compute directly the probability that at least two colours appear, which leads to complicated casework. Another mistake is to compute probabilities for only one colour and forget to sum over all colours. Using the complement method simplifies the logic and reduces errors.

Final Answer:
The probability that the four pearls drawn are not all of the same colour is 90/91, which corresponds to Option D.