Coin Game

The Problem

You and your opponent each have a coin. In each round you each choose to show either the heads or tails side of the coin. If both coins show heads, your opponent pays you $3. If both show tails, she pays you $1. If they don't match, you pay her $2. Is this a fair game?

Note: This is only a demonstration. We are not playing for real money.

Playing the Game

Your coin and your opponents coin are shown in the window. For each round of play, click either the Heads button or the Tails button to indicate which side you want to show to your opponent. (You can also use the Alt-H key for Heads and Alt-T for Tails.) Your opponent will make an independent choice for her coin.

You can automatically play multiple rounds. Enter the number or rounds in the field to the right of the coins and click the Auto Play button. The program automatically plays the number of rounds you specify. It will alternate between heads and tails for your coin.

The bottom of the screen shows the results of the most recent round and a summary of the play so far. You can choose to see just results for the current session or to see historical results (since about June 1, 2003).

The Analysis

If you each play randomly, then the game is even. Out of four random rounds you can expect one case of Heads-Heads (+$3), one case of Tails-Tails (+$1), one case of Heads-Tails (-$2) and one case of Tails-Heads (-$2). This nets out to $0. However, your opponent can skew the results by playing heads or tails more often. In the following chart, the horizontal axis shows the percent that your opponent plays tails. The red line shows your winnings per round when you play heads; the blue line shows your winnings per round when you play tails.

If your opponent plays tails less than 60% of the time, you have the advantage when you play heads (red line above the axis). If your opponent plays tails greater than two-thirds (66.6%) of the time, you have the advantage when you play tails (blue line above the axis). Note specifically that if your opponent plays tails more than 60% and less than 66.6% of the time, your opponent has the advantage whether you play heads or tails (both lines below the axis).

In this computer version of the game, your opponent plays 62.5% tails. Over time, you should lose an average of about $1 every 8 rounds. However, because your opponent's advantage is very small, you might find you come out ahead even after dozens of rounds.

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