This is the question: Suppose we write the numbers

1,2,3,4,5,6, and then color each number red or blue. The coloring is called factorific if there is at least one blue number, and for each blue number, all of its divisors are also blue. Grogg randomly colors some, all, or none of the numbers from 1 to 6 blue, and he colors the rest red. What is the probability that his coloring is factorific?



My answer: 1/4. Is this correct? please don't link to other questions, just tell me if it's right or wrong.




 Jul 13, 2019

I got \(\dfrac{1}{8}\), but I am not so sure either. Will wait for someone else to answer.

 Jul 13, 2019

I'm seeing 1 valid arrangements using 1 blue {1}

5 valid arrangements using 2 blues {1x}, x=2-5

2 valid arrangements using 3 blues {124},{136}


and that's it


8 valid arrangements, 26 = 64 possible arrangements


P[valid]=8/64 = 1/8

Rom  Jul 13, 2019

youre actually right.......!

 Jul 13, 2019

This is an AoPS question and you’ve not demonstrated any solution method for your answer.   

Why the opposition to linking to other questions? Did the swill from jollyjellyjojo nauseate you?

According to him, the official answer from AoPS is (1/4).  With a name like jollyjellyjojo how could you go wrong? W T F 


My solution is in this link.  https://web2.0calc.com/questions/another-one-thanks

(The nauseating swill from jollyjellyjojo is also in this link, so you may want to take a double dose of scopolamine)

I demonstrate that repeating Grogg’s process produces a binomial distribution.   


It’s unclear (to me) of how MaxWong and Rom arrived at (1/8). Melody’s answer is here: https://web2.0calc.com/questions/help_47515 She makes a clear assumption in her solution that 50% of the numbers are randomly colored blue.  They are all excellent mathematicians, so ... 


I’ve not changed my position that the verbs in this question present a binomial selection process, not a uniform random selection process.  So it would seem this is a question of semantics: one of how to translate the verbs of English into mathematics.


You have three choices: (1/8), (1/4), and (1/3).   Just pick your semantics and your number. Good Blŏŏdy Luck




 Jul 13, 2019

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