Geometry

Question Analysis
This question involves calculating the probability of three successive dependent events.
The main focus is on extending the concept of dependent - event probability from two events to three events and correctly applying the multiplication rule for each step.

Key Concept Explanation
For multiple dependent events, the outcome of each previous event affects the probabilities of the subsequent events.
As each ball is drawn, the total number of balls and the number of balls of each color change, which in turn affects the probability of drawing a ball of a particular color in the next draw.

Step-by-Step Solution
Calculate the probability of drawing a purple ball on the first draw:
There are 15 balls in total, and 6 are purple.
So, .
Calculate the probability of drawing an orange ball on the second draw given that a purple ball was drawn on the first draw:
After removing one purple ball, there are 14 balls left, and 5 are orange.
So, .
Calculate the probability of drawing a yellow ball on the third draw given that a purple ball was drawn first and an orange ball was drawn second:
After removing a purple and an orange ball, there are 13 balls left, and 4 are yellow.
So,

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