It is all part of a grand pattern that can be called the laws of chance or the laws of probability. In some situations it is easy to figure out the chance of a particular series of events occurring. In other situations, it requires an expert with vast resources at his command.
Let’s make a minor digression and deal with dice as a simple exercise in probabilities. If you have no interest in this, just skip the next few paragraphs. There is no skill at all in the tossing of honest dice, and the information which follows can be used to compute the result of almost any dice problem.
If a die is perfect, its chance of coming to rest on a particular side is one in six. In tossing a pair of dice, there are 36 (6 x 6) possible combinations. Each single combination has an equal chance of coming up on each toss. The odds against a particular combination are 35 to 1. Here is our problem. Is it an even bet that the dice will pass?
The 36 possible combinations of a pair of dice are all listed in the first column. They are grouped according to the results that have a meaning in a dice game. These results, the total value, are shown in the second column. The third column is the count of the number of combinations in the first column that yield the value in the second column. There are four combinations that make a five, six combinations that make a seven, and so on. Remember that each of the 36 combinations has an equal chance of occurring.
The fourth column gives the true odds for one roll of the dice. It is 11 to 1 that you don’t throw a four; it is 5 to 1 that you don’t throw a seven; etc. In a Nevada casino, if you bet on the Field Numbers, you have 2, 3, 4, 9, 10, 11, and 12. The house has 5, 6, 7, and 8. You have seven numbers and the house has only four. Consider, however, the number of combinations. The house has twenty. You have sixteen. You bet even money when you should be getting 5-to-4 odds. Every nine times you bet a dollar on the “Field” you should have a net loss of $1.
In the fifth and sixth columns we have assigned the Pass and Don’t Pass values. We shall study the problem in terms of the 36 sets of combinations. In dice, 7 or 11 on the first roll win immediately. Since 7 should come up six times out of 36 rolls, we enter 6 in the Pass column. Eleven comes up twice in 36, so we enter 2 in the Pass column. Don’t Pass; get the 2, 3, and 12 on the first roll. If a 4, 5, 6, 8, 9 or 10 come up on your first roll, that number is your point. On succeeding rolls, if your point comes up before a 7, you win. If a 7 comes up before your point, you lose.
Let’s review these prospects. A 4 has three combinations. A 7 has six combinations. Each of these combinations has an equal chance of coming up. So the odds are 2 to 1 against passing. In three chances, you should pass once and lose twice. So we enter 1 in the Pass column and 2 in the Don’t Pass column. The odds are 3 to 2 against a 5 coming up before a 7. The dice should pass two out of five. However, as 5 came up only four times in our initial set of 36, the number of winners is 4 x 2/5 or 8/5. The rest of columns 5 and 6 are filled out by the same type of arithmetic. The fractions convert to decimals as indicated in the parentheses.
The answer to our problem is obtained by adding up the values in columns 5 and 6. The result can be described in various ways, but in any way it means that honest dice will pass less often than they will not pass. Here is the simplest way of describing the result. The odds are approximately 1.03 to 1 against the dice passing. A mathematician would say that the probability the dice will pass is .493.
