Ponder This Challenge - February 2012 - Biased coin game

Ponder This Challenge:

Thank you, Kipp Johnson, for the wonderful challenge:

To decide who would play first in a human vs. computer game, the human suggested flipping a coin it produced from its pocket, stating “heads you win, tails I win.” The computer replied, “Humans are very devious. I know that your coin is not fair, and that the probability of heads on that coin is slightly less than one half. To make this perfectly fair, flip the coin repeatedly, and I win if the coin first shows heads on toss number 1, 14, 15, 19, 20, or 23. You win if it first shows heads on any other toss.”

If the computer is correct and p is the probability that the coin shows heads, give the best rational approximation to p with a denominator having fewer than 10 digits.

Bonus question: What is the significance of the six positions the computer chose?

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Solution

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  Letting q=1-p, the probability that a heads will first appear on one of those six tosses is p(1+q^13+q^14+q^18+q^19+q^22). To make the tossing fair, this probability must equal 1/2. Solving the equation gives only one real solutions: p = 0.499905242928506776611191007553...

  Using a continued fraction, we find the best rational approximations to be 94,668,920 /189,373,729.

  The answer to the bonus question is that the six numbers, interpreted as positions in the alphabet, give the letters that make up the word WATSON.

Solvers

• * (02/01/2012 01:57 PM EDT)
• Deron Stewart & Deane Stewart (02/01/2012 01:57 PM EDT)
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