Geometry

Question Analysis
This question involves calculating the probability of three successive dependent events of drawing colored pencils of specific colors from a box.
The main focus is on correctly identifying the number of pencils of each color at each draw and applying the multiplication rule for dependent events to find the combined probability of the three events occurring in the given order.

Key Concept Explanation
Dependent events imply that the result of each previous pencil draw changes the composition of the pencils in the box, thereby affecting the probabilities of the subsequent draws.

Step-by-Step Solution
Calculate the probability of drawing a blue pencil on the first draw:
There are 4 blue pencils out of 15 total pencils.
So, .
Calculate the probability of drawing a green pencil on the second draw given that a blue pencil was drawn on the first draw:
After removing one blue pencil, there are 14 pencils left, and 3 of them are green.
So, .
Calculate the probability of drawing a purple pencil on the third draw given that a blue pencil was drawn first and a green pencil was drawn second:
After removing a blue and a green pencil, there are 13 pencils left, and 1 of them is purple.
So,

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