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Puzzle 2, The Ridgefield Firemen’s Fair

Tickle your brain. Fire up those neurons and see if you can solve the puzzle.

Every year the Ridgefield Volunteer Firemen host a fair at ERMS. They try to raise money to fund their department.  Good cause.  How can we help?

Well, what if they did this, what if they introduced a new gambling game where they flip a coin. If it lands on heads, the firemen pay you $1. If it lands on tails, you must pay the firemen $1. Sounds fair. But therein lies the problem, how can the firemen make money on this game? It’s not really possible exactly because it is a fair game.

So what can we do to mix things up a bit?

What if we changed the game slightly. What if there were two coins. You get to flip one coin and the firemen get to flip the other coin.  If both coins are heads, you get paid $3.  If both coins are tails, you get paid $1.  If the coins turn out to be a heads and tails, or a tails and heads, then you must pay the firemen $2.  That sounds fair again right? There are four options, you can earn $3 or $1, the firemen can earn $2 or $2.

And let’s add one little catch.  Instead of flipping the coins. You and the firemen get to choose your coin, meaning you can look at your coin and set it to either heads or tails and on the count of three you lay the coin down on the table. Still sounds fair doesn’t it, each side gets to choose what they want on their coin.

So the puzzle is this, how can this game make money for the firemen?

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Zen Sailor February 11, 2013 at 03:37 PM
because people are basically fair and generous.....that,and the fact that the fireman probably won't have to send any goons out to collect.
Schrdinger's Cat February 20, 2013 at 06:17 AM
It’s really interesting how this one plays out. If you can understand this, you can make a fortune on Wall Street. I’ll set up some variables: Pfh = the Probability the fireman chooses heads. Pft = the Probability the fireman chooses tails. Pyh = the Probability you choose heads. Pyt = the Probability you choose tails. What do kind of payout can be expected? The payout for you is the probability of you choosing while at the same time the fireman chooses heads. If it happened, you’d get $3. I could write an equation for that: Payout = $3 * Pfh * Pyh. You’d also make $1 for a tails tails situation. Payout = $1 * Pft * Pyt The fireman on the other hand would get a payout for: Payout = $2 * Pft * Pyh and Payout = $2 * Pfh * Pyt From the fireman point of view, his total combined payout is: Total Payout = (-$3* Pfh * Pyh) + (-$1 * Pft * Pyt) + ($2 * Pft * Pyh) + ($2 * Pfh * Pyt)
Schrdinger's Cat February 20, 2013 at 06:18 AM
But we also know that a player must choose either heads or tails 100% of the time. So we can say: Pfh + Pft = 1 or Pfh = 1 - Pft We can substitute that into the larger equation and get: Total Payout = (-$3* Pfh * Pyh) + (-$1 * (1 - Pfh) * (1 -Pyh)) + ($2 * (1 - Pfh) * Pyh) + ($2 * Pfh * (1 - Pyh)) Simplifying: Total Payout = -8 * Pfh * Pyh -1 + 3 * Pfh + 3 * Pyh Looks boring, but watch this. If the fireman intentionally chose heads 3/8ths of the time, then you get this: Total Payout = -8 * 3/8 * Pyh - 1 + 3 * 3/8 + 3 Pyh The Pyh drops out of the equation. It doesn’t matter what you choose for you coin. Total Payout = 1/8 This is a positive number. It’s $0.12. It means that on average, if the fireman chooses heads 3/8ths of the time, he will come out ahead by $0.12 on every turn. Play 1000 times per night and the fireman will make $125. A very unexpected result, that a game involving heads and tails can give one player an advantage. Free money! I look forward to seeing this game at next year’s Firemen’s Fair.

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