In explaining how to play craps, I’ve always tried to tell people to keep in mind that odds and probability are two completely separate and very different things.  Usually people are too busy trying to wrap their heads around the basic rules of the game to even start on the “odds vs probability” concept.  I’ve always wished there was a better, but similar, example that didn’t distract people with thoughts like “What’s a field bet?”  Then I came across this stuff about Yahtzee on the internet which seems to cover the same concept.

So, since far more people are familiar with Yahtzee^*:

A Yahtzee, as most people who’ve played the game know, is when all five dice have a matching number.  The odds are 1 in 1296 of rolling a Yahtzee in one toss of all five dice.  If (when) you miss, you were to roll all five dice repeatedly over and over until you rolled a Yahtzee, each roll would increase your probability of rolling a Yahtzee by just a little bit.

However,  the probability of your rolling a Yahtzee doesn’t change in matching fashion with your number of rolls in relation to the 1296.  In other words if you’ve rolled 100 times, probability doesn’t dictate that you’ll definitely roll a Yahtzee within the next 1196 rolls.  Oh yeah, and your odds never change.

Your odds of rolling a Yahtzee are 1 in 1296 with each and every roll of the dice.  Dice are indifferent to the results of the previous roll.  Because of this indifference, probability diverges from odds and you don’t have a 100% probability of hitting the Yahtzee within the 1296 rolls.  It has to do with exponentiation, a concept I’m not going to explain in this blog post because those of you who get it are already ahead of me and those that don’t will stop reading in frustration.  In fact, I barely have a grasp on it myself.

You have a 63% of rolling a Yahtzee (37% chance of failure) in 1296 rolls. Within 2920 rolls you have about a 90% probability of rolling a Yahtzee. Odds and probability don’t match up mathematically except on the very first roll (and with zero rolls but that’s a whole different matter altogether).  Ain’t that a bummer?

This concept applies to craps as well, however the percentages of success/failure (rolling a seven) are a much different being that there are only two dice but six number combinations involved (vs five dice and six combinations for Yahtzee). In craps, the probability curve is a lot tighter and you’re fairly likely to roll the seven within seven rolls. In fact, on the third roll, the percentage is close to 52%. On the fourth, it’s about 77%.

Seven is the most common number rolled because it’s the most common combination on two dice. There are six ways to get a seven and only four ways to get a five. The odds (and first roll probability) of rolling a five are 4 in 36, which is going up against “seven out seven” odds of 6 in 36. While the probability of rolling a five increases with each roll, the probability of rolling a seven increases at a greater rate, for example (4/36)³ for a five vs (6/36)³ for a seven in three rolls.

The thing about this probability calculation, however, is that it’s an empirical measurement using a finite number of rolls. For example, the probability of not rolling a seven in craps within three rolls is (5/6)³. However, the finite number of rolls is not predetermined in craps, it’s determined by the occurrence of the result (“seven out seven”). Because of this, before the seven is rolled, probability is more abstract than empirical.

I’ve lost you, haven’t I?

OK, just try to keep in mind that odds and probability are not the same thing. Each time you roll the dice, your odds of getting a seven are the same. Each time you roll the dice after a not rolling a seven, the probability of hitting seven increases a little bit. Fairly quickly, if a seven is not rolled, the probability of rolling a seven will far surpass the odds and almost dictate it’s happening.

Oh yeah, then there’s the randomness of the statistical outliers (a “hot roller”) that make you feel virtually immortal and start taking silly risks.  Doubled your money at a hot table?  It’s called “probability” for a reason. Yeah… walk away while you’re still up and hit the buffet.
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^*By the way, this is meant to be more humorous than informational. 
Of course, to those who truly know me, this is obvious.

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