"Your artistic talents win the approval and applause of others."

"Lucky numbers 33, 21, 47, 5, 28, 49."

Being a geek, I found the prime factors of my lucky numbers:

lucky number |
prime factors |

33 |
11, 3 |

21 |
7, 3 |

47 |
(prime) |

5 |
(prime) |

28 |
7, 2 |

49 |
7 |

Perhaps the universe is trying to tell me something with the preponderence of sevens. What are the chances of seven turning up three times in prime factors of six randomly chosen numbers? (Note that a reasonably broad survey of Dragon Chef fortune cookies suggests that numbers over 99 are all unlucky.)

Occurence of prime factors in numbers under 100:

prime factors |
occurences |
probability of one occurence |

2 |
49 |
0.49 |

3 |
33 |
0.33 |

5 |
19 |
0.19 |

7 |
14 |
0.14 |

11 |
9 |
0.09 |

(Table truncated for brevity. Numbers omitted: 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.)

The chance of getting precisely one permutation is

(0.14)(0.14)(0.14)(0.86)(0.86)(0.86)

or 0.001745,

but any grouping of numbers with prime factors of seven would have seemed equally significant. Unsure of how to proceed, I turned to my designated hitter.

Suze: Hey honey, I have a question for you.

Mike: Okay ...

Suze: If I have a box with ten blue balls and ninety red ones, what are the chances of getting exactly three blue balls if I choose six?

Mike: Do you care about the ordering?

Suze: No.

Mike: So the chances of getting a single ordering are (0.9)(0.9)(0.90(0.1)(0.1)(0.1). But then you have to account for the combinations. *(A meandering discussion ensues, and then ...)* It probably involves factorials.

Suze: That makes sense.

Mike: What is this for, anyway?

Suze: It's so stupid that you are not going to believe it.

Mike: *(waits for explanation)*

Suze: See, I opened this fortune cookie, looked at the lucky numbers, factored them into prime numbers, and there seemed to be lots of sevens, so I'm trying to figure out if the universe is giving an opinion on what I should be knitting.

Mike: It's for astrology.

Suze: Well yes.

Mike: I would just write a computer program.

Suze: The brute force calculation would take me less time.

Mike: Just tell me the paramaters and I'll do it.

Suze: Thanks anyway, but I think I'll look for the equations.

After noodling around on the web, I came up with a transparent reference, then this:

C(n, r) = n! / r! * (n-r)!

where

- C = number of combinations (the number I want)
- N = number in group (lucky numbers, in this case 6)
- R = number in subgroup (occurence of lucky numbers with a prime factor of seven, in this case 3)

Plugging in the numbers gives

C = 6! / 3! x (6 - 3)!

C = (6 x 5 x 4 x 3 x 2 x 1) / (3 x 2 x 1) x (3 x 2 x 1)

= 720 / 6 x 6

= 20

And the probability of getting three prime factors of seven in six lucky numbers should be the probability of a single permutation times the number of combinations:

P = 20 x 0.001745, or

= 0.0349,

or roughly one chance in 28.

Having slogged through this, I then asked what the chances are of choosing two primes among six lucky numbers. Chance of a single combination is

(0.16)(0.16)(0.84)(0.84)(0.84)(0.84)

or 0.01275,

and the number of combinations is

C = 6! / 2! x (6 - 2)!

C = (6 x 5 x 4 x 3 x 2 x 1) / (2 x 1) x (4 x 3 x 2 x 1)

= 720 / 2 x 24

= 15.

So the chance of choosing two primes in six numbers is

P = 15 * 0.01275

= 0.1912

or roughly one chance in five.

So the universe it suggesting that a new project based on sevens would be well received. Alternately, the universe is telling me that people liked the Circus Blanket, which was based on three permutations of seven. Damn astrology.

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