My Birthday Problem (hard)?

Question

If the probability of a birthday cake candle being defective is a constant, "p", with all candles independent, then the probability of a candle problem on my birthday cake at MY age can be written as a product of irreducible integral polynomials in "p", the largest of which has degree 42.  How old am I?  

(That is: irreducible over the integers.)

gianlino · Accepted Answer

Not so hard if one follows links you gave elsewhere.

http://en.wikipedia.org/wiki/Cyclotomic_polynomial

You need to know which numbers have 42 roots of unity. 

49 is one and 98 is another. So if you decide only one factor has maximal degree it must be 49. Otherwise 98 is also ok.

QC · Answer

49

n = number of candles = age

P(you have candle problem)
= 1 - P(you don't have candle problem)
= 1 - P(all candles work)
= 1 - (1-p)^n

When n = 49, we get:

1 - (1-p)^49 = 

p (p^6 - 7p^5 + 21p^4 - 35p^3 + 35p^2 - 21p + 7) (p^42 - 42p^41 + 861p^40 - 11480p^39 + 111930p^38 - 850668p^37 + 5245786p^36 - 26978329p^35 + 118030220p^34 - 445892405p^33 + 1471449518p^32 - 4280613736p^31 + 11058441520p^30 - 25520354440p^29 + 52866953601p^28 - 98695963464p^27 + 166580329440p^26 - 254845509828p^25 + 354114367425p^24 - 447609860880p^23 + 515268321960p^22 - 540579017881p^21 + 517042658706p^20 - 450842146860p^19 + 358247813140p^18 - 259220974971p^17 + 170600092656p^16 - 101957867200p^15 + 55220421361p^14 - 27032714744p^13 + 11923284464p^12 - 4719614536p^11 + 1668499196p^10 - 523702102p^9 + 144880603p^8 - 35006601p^7 + 7302960p^6 - 1295952p^5 + 191786p^4 - 23030p^3 + 2156p^2 - 147p + 7)

However, I haven't exactly figured out how to get result without knowing answer is 49 :(

natalie · Answer

I really hate stats. I tried, filled up a whole page of scribbles and gave up. You are the maths master.

anniec · Answer

omg i tried

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My Birthday Problem (hard)?

4 Answers