The solution to this question requires analysis of mutually exclusive (conditional) sets. Each set has an individual probability. The sum of these probabilities times the probability of choosing one of the sets determines the overall probability of a “factorific” coloring.

From the question: The condition requires all divisors of a blue number to be blue. Note a number can be blue __without__ it being a divisor of all greater numbers __as long as its divisors are blue__.

*Grogg chooses to color **none** **(0)** of the numbers.

__Probability of factorific__: Zero (0)

*Grogg chooses to color **one (1) **of the six (6) numbers.

__Probability of colorific__: **(1/6).**

__Explanation:__ Number one (1) is the only positive integer that has one divisor.

*Grogg chooses to color **two** **(2)** of the six (6) numbers.

__Probability of factorific__: = **(1/5)**

__Explanation:__ Number of ways to choose two numbers from 6 numbers = (nCr(6, 2)) = 15.

Number one (1) must be in the set because it’s a divisor of all integers. The other number has to be a prime number (2, 3, or 5)

{1, 2}

{1, 3}

{1, 5}

Number of sets that have a one (1) and a prime: Three (3).

Probability (3/15) = (1/5)

*Grogg chooses to color **three (3)** of the six (6) numbers.

__Probability of factorific__: (5/20) = **(1/5)**

__Explanation:__ Number of ways to choose 3 numbers from 6 numbers = (nCr(6, 3)) = 20

Number one (1) must be in the set because it’s a divisor of all integers.

These 4 sets meet the conditions:

{1, 2, 3} Two (2) is blue here, it’s not a divisor of three, but one (1) is its divisor, so this is valid.

{1, 2, 4} all divisors are blue for all blue numbers.

{1, 2, 5} ‘’

{1, 3, 5} ‘’

~~{1, 5, 6} ‘’~~

Five of 20 sets meet the conditions: (1/5)

*Grogg chooses to color **four (4)** of the 6 numbers.

__Probability of factorific__: **(4/15)**

__Explanation:__ Number of ways to choose 4 numbers from 6 numbers = (nCr(6, 4)) = 15

Number one (1) must be in the set because it’s a divisor of all integers.

These 3 sets meet the conditions

{1, 2, 3, 4} all divisors are blue for all blue numbers.

{1, 2, 3, 5} ‘’

{1, 2, 3, 6} ‘’

{1, 2, 4, 5} ‘’

Four of 15 sets meet the conditions: (4/15)

*Grogg chooses to color **five** **(5)** of the six (6) numbers.

__Probability of factorific__: **(1/2)**

__Explanation:__ Number of ways to choose 5 numbers from 6 numbers = (nCr(6, 5)) = 6

Number one (1) must be in the set because it’s a divisor of all integers. Number of sets of five (5) that have one (1) as an element = (nCr(5, 4)) = 5.

Three sets meet the conditions

{1,2,3,4,5} all divisors are blue for all blue numbers.

{1,2,3,4,6} ‘’

{1,2,3,5,6} ‘’

Three sets of 6 sets meet the conditions (3/6) = (1/2)

*Grogg chooses to color six of the six (6) numbers.

__Probability of factorific__: **(1) or 100%**

Explanation:All numbers are blue and all numbers have all their divisors colored blue.

---------

Sum of individual (mutually exclusive) probabilities: ((0)+(1/6)+(1/5)+(1/4)+(4/15)+(1/2)+(1))

Grogg has a (1/7) probability of picking one of the seven sets (including the empty set).

~~(1/7)*((0)+(1/6)+(1/5)+(1/4)+(4/15)+(1/2)+(1)) = (143/420) ~~**≈ **34.05**%**

(1/7)*((0)+(1/6)+(1/5)+(1/5)+(4/15)+(1/2)+(1)) = (143/420) ≈ 33.33%

The overall probability that Grogg’s coloring is factorific is **≈ **__33.33%__

**Sources: A genetically enhanced chimp brain, and comprehensive programming of basic set theory from Lancelot Link.**

GA

Edit: Corrected error