fortune cookie

"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.

Comments

[IHateToast]

was just watching the abc (like pbs, but aussie) on the muslim years in europe. there was a big section on alhambra (i'm finishing a bag during this, so lots of details are lost when i had to concentrate) and one scholar noticed the proportion of all the rectangles. if you take the one square in the building and its diagonal, you have the height of the next rectangle. that one's diagonal is the height of the next and so on. very cool. nice that alhambra was left in peace unlike the moors, their papers, books, etc when catholicism returned. aaah angry, self righteous people. my favourite.

math is beautiful, it's just often taught in an ugly manner.

[Liz (the crazed weasel)]

hee, hee! Always happy to see a use for factorials that doesn't really involve Taylor series in any way because I hate Taylor series, yes I do do do.

I keep trying to tell my kids, "Math is Beautiful! it is Exciting! it is Pretty and Cool!" and they go, yeah mom but you also eat calamari. (end of debate) (because clearly the opinions of a person who thinks squid are food are suspect)

[Heather]

NICE! Although I disapprove of spark notes on principle, I like the results. (And since you used them in the intended application)...I just couldn't get why the universe would give you approval and applesauce (I read it too fast!) ;-)

[Rebecca H.]

Can't say that I followed any of that, other than "I would just write a computer program." Because that is infact just the sort of thing my Mike would say if I were in a cold cold hell ever talk about numbers in that way ;)

[Jenn C.]

oh, that makes my head hurt on a not-enough-sleep morning.

I love how much you revel in your geeky self.

[berlinBat]

Hilarious! Thanks for making my day :)

[The Feminist Mafia]

What Colleen said.

And ... "ow, my head hurts."

Thank god there are math/science women like you in the world. Very impressive, my dear. Very impressive.

[Danielle]

Hmmm. I think the most informed comment that I can leave on this post is "hey, I liked the circus blanket too!" But props to you for the mathematical reasoning! (If I remembered more of the math that I learned years ago, I probably would have calculated it too!)

[colleen]

Can I say "wow"? Wow!