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 Features
Statistically Speaking, It's Probably a Good Game, Part 1: Probability for Game Designers
 The Law of Averages Got Vetoed
Mistake #3 (from back before Scene 24b) is a similar, but extended error: believing that over the long run, everything will “even out” — the Law of Averages. It’s true that, out of 1000 flips of a coin, you’d expect to see roughly 50% heads and 50% tails. But there is no such thing as a “correction.” If you flip a coin ten times and get 8 heads against 2 tails, there is no global essence or power that is going to squeeze a few more tails into the next 10 flips. You would be making a grave philosophical error to assume that “tails are due”, and an even graver error to put big money on it. Peter Webb has an excellent short discussion on this subject at his website (see recommended reading at the end of this article).
The gist is, if you flip a coin 1 million times, you’ll expect the heads and tails split to be close to 50%. But don’t expect the NUMBER of heads flips to equal the NUMBER of tails flips — in fact, it’s very likely that they will be off by hundreds or even thousands. Remember, you could have 10,000 less heads than tails and the division would still be very close to 50%/50% (49%/51%, to be exact). So don’t put money on assuming that an 8-to-2 heads lead (+6 heads) will be corrected as you flip more coins! It’s very likely that even if the heads/tails split will be close to 50%/50% in the long run, the actual difference between number of heads and number of tails will probably grow as the total number of flipped coins grows.
Converse Probability
It’s easy to find formulas that will help you calculate the chances of independent or related events. Sometimes, though, it can be very, very difficult to calculate more involved probabilities. One trick that you can pull out of your hat to save the day is the concept of “converse probability.” To calculate converse probability, instead of trying to determine the chances that something will happen, you instead calculate the chances that something won’t. Then, you subtract this number from 1.0 (100%) to get the probability that you are looking for.
Converse Prob 101: Easy Example
You are about to roll a six-sided die. What are the chances that you’ll roll a “6”? Although we already know the answer, we’ll use converse probability to verify it. The chances you won’t roll a “6” are 5/6 (5 out of 6 of the die sides are not a 6). Therefore, the chance of rolling a “6” is 1 – 5/6 = 1/6, or the familiar 16.7%. In other words, if you won’t roll a “6” 5 out of 6 times, then you will roll a “6” 1 out of 6 times. That almost makes sense!
Converse Prob 201: Flush with Excitement
Here’s a situation where converse probability really is a money saver. It’s Texas Hold’em, and you are four cards to a heart flush with two cards to come. In other words, if a heart comes on the Turn or the River, you’ll complete your flush**. What are the chances this will happen?
**Given the choice between the two, I recommend completing your flushes on the River — this has the greatest chance of psychologically destroying your opponents and sending them into apoplectic fits. Nobody likes getting Rivered.
It’s very easy to calculate the chances of a heart coming on the next card. There are 9 hearts remaining “in the deck” that have not been flipped up yet (13 to begin with minus the 4 already showing between the flop and your hand). There are 47 cards total in the deck (52 minus the two in your hand and the three flopped on the board). Therefore, the chances of flipping a heart on the next card are 9 out of 47, or 9/47. If that card isn’t a heart, then the chance of flipping a heart on the following card is 9/46 (there are still 9 hearts remaining, but one less card total in the deck).
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