title: Efficient Method for Calculating the Probability of a Set of Outcomes?
body: <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>