IBM Ponder This #166 · February 2012

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?

Solution

Best opened after a real attempt

To be added.