Humuhumunukunukuapua'a

Probability Level 5

The state fish of Hawaii is a trigger fish known by it's Hawaiian name:

Humuhumunukunukuapua’a \text{Humuhumunukunukuapua'a}

Humuhumunukunukuapua'a Humuhumunukunukuapua'a

So, let's suppose, Hawaii decided that 21 letters is far too long of a name for a state fish. So every year they decide to shorten it by one letter which they will knock off the beginning or the end, decided by the flip of a coin.

So, for example, after the first year it would be one of the following:

  • Umuhumunukunukuapua'a (If they knock off the first letter)
  • Humuhumunukunukuapua (If they knock off the last letter)

When the fish's name is down to only one letter, the probability that it will called "N" is a b \dfrac{a}{b} , where a a and b b are coprime positive integers.

What is a + b a + b ?

Image credit: http://www.animalpicturesociety.com


The answer is 325129.

Geoff Pilling by Geoff Pilling , 53, USA

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2 solutions

Geoff Pilling
Feb 12, 2017

For each letter in position n n for a word of length N N , the probability that this letter will be the remaining letter at the end is given by:

P = ( N − 1 n − 1 ) 2 N − 1 P = \dfrac{\binom{N-1}{n-1}}{2^{N-1}}

The letter "N" appears in positions 9 9 and 13 13 . So the probability that "N" will be the final name is:

Probability("N") = ( 21 − 1 9 − 1 ) 2 21 − 1 + ( 21 − 1 13 − 1 ) 2 21 − 1 = \dfrac{\binom{21-1}{9-1}}{2^{21-1}} + \dfrac{\binom{21-1}{13-1}}{2^{21-1}}

= 125970 1048576 + 125970 1048576 = \dfrac{125970}{1048576} + \dfrac{125970}{1048576}

= 62985 262144 = \dfrac{62985}{262144}

62985 + 262144 = 325129 62985 + 262144 = \boxed{325129}

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Ashish Gupta
Feb 12, 2017

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice approach, Ashish!

Geoff Pilling - 4 years, 3 months ago

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Thanks. But nothing fancy here. Same as your solution. I could never remember the difference between negative binomial distribution and hyper-geometric distribution back at school. I always used to count by arranging O's and X's. :)

Ashish Gupta - 4 years, 3 months ago

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