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POEfficient Method for Calculating the Probability of a Set of Outcomes?
 

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  1. POEfficient Method for Calculating the Probability of a Set of Outcomes?
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    <p>Let's say I'm playing 10 different games. For each game, I know the probability of winning, the probability of tying, and the probability of losing (each game has different probabilities).</p> <p>From these values, I can calculate the probability of winning X games, the probability of losing X games, and the probability of tying X games (for X = 0 to 10).</p> <p>I'm just trying to figure out the probability of winning <em>W</em> games, tying <em>T</em> games, and losing <em>L</em> games after playing all 10 games... and hopefully do better than O(3^n). For example, what is the probability of winning 7, losing 2, and tying 1?</p> <p>Any ideas? Thanks!</p> <hr> <p>Edit - here's some example data for if there were only 2 games:</p> <p>Game 1:</p> <ul> <li>win: 23.3%</li> <li>tie: 1.1%</li> <li>lose: 75.6%</li> </ul> <p>Game 2:</p> <ul> <li>win: 29.5%</li> <li>tie: 3.2%</li> <li>lose: 67.3%</li> </ul> <p>Based on this, we can calculate the probability after playing 2 games of:</p> <hr> <ul> <li>0 wins: 54.0%</li> <li>1 win: 39.1%</li> <li>2 wins: 6.9%</li> </ul> <hr> <ul> <li>0 ties: 95.8%</li> <li>1 tie: 4.2%</li> <li>2 ties: 0.0%</li> </ul> <hr> <ul> <li>0 losses: 8.0%</li> <li>1 loss: 41.1%</li> <li>2 losses: 50.9%</li> </ul> <hr> <p>Based on these numbers, is there a generic formula for finding the probability of <em>W</em> wins, <em>T</em> ties, and <em>L</em> losses? The possible outcomes (W-L-T) would be:</p> <ul> <li>2-0-0</li> <li>1-1-0</li> <li>1-0-1</li> <li>0-1-1</li> <li>0-2-0</li> <li>0-0-2</li> </ul>
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    1. COWell, can you tell us if the games _are_ independent? If not, how do they depend on each other?
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    2. COThe games ARE independent.
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    3. CO@Kenny. OK, see my answer. Hope it helps!
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