Hack 43. Play with Dice and Get Lucky

Here are some honest wagers using honest dice. Just because you aren't cheating, though, doesn't mean you won't win.

It is an unfortunate stereotype that statisticians are glasses-wearing introspective nerds who never have a beer with the gang. This is such an absurd belief, that just thinking about it last Saturday and Sunday at my weekly Dungeons & Dragons gathering, I laughed so hard that my monocle almost landed in my sherry.

The truth is that displaying knowledge of simple probabilities in a bar can be quite entertaining for the patrons and make you the life of the party. At least, that's what happens according to my Uncle Frank, who for years has used his stats skills to win free drinks and pickled eggs (or whatever those things are in that big jar that are always displayed in the bars I see on TV).

Here are a few ways to win a bet using any fair pair of dice.

Distribution of Dice Outcomes

First, let's get acquainted with the possibilities of two dice rolled once. You'll recall that most dice have six sides (my fantasy role-playing friends and I call these six-sided dice) and that the values range from 1 to 6 on each cube.

Calculating the possible outcomes is a matter of listing and counting them. Figure 4-2 shows all possible outcomes for rolling two dice.

Figure 4-2. Possible outcomes for two dice

This distribution results in the frequencies shown in Table 4-15.

Table Frequency of outcomes for rolling two dice

Total roll	Chances	Frequency
2	1	2.8 percent
3	2	5.6 percent
4	3	8.3 percent
5	4	11.1 percent
6	5	13.9 percent
7	6	16.7 percent
8	5	13.9 percent
9	4	11.1 percent
10	3	8.3 percent
11	2	5.6 percent
12	1	2.8 percent
Total number of possible outcomes	36	100 percent

The game of craps, of course, is based entirely on these expected frequencies. Some interesting wagers might come to mind as you look at this frequency distribution. For example, while a 7 is the most common roll and many people know this, it is only slightly more likely to come up than a 6 or 8.

In fact, if you didn't have to be specific, you could wager that a 6 or an 8 will come up before a 7 does. Of all totals that could be showing when those dice are done rolling, more than one-fourth of the time (about 28 percent) the dice will total 6 or 8. This is substantially more likely than a 7, which comes up only one-sixth of the time.

Bar Bets with Dice

My Uncle Frank used to bet any dull-witted patron that he would roll a 5 or a 9 before the patron rolled a 7. Uncle Frank won 8 out of 14 times.

Sometimes, old Frankie would wager that on any one roll of a pair of dice, there would be a 6 or a 1 showing. Though, at first thought, there would seem to be at least a less than 50 percent chance of this happening, the truth is that a 1 or 6 will be showing about 56 percent of the time. This is the same probability for any two different numbers, by the way, so you could use an attractive stranger's birthday to pick the digits and maybe start a conversation, which could lead to marriage, children, or both.

If you are more honest than my Uncle Frank (and there is a 98 percent chance that you are), here are some even-money bets with dice. The outcomes in column A are equally as likely to occur as the outcomes in column B:

A	B
2 or 12	3
2, 3, or 4	7
5, 6, or 7	8, 9, 10, 11, or 12

The odds are even for either outcome.

Why It Works

For the bets presented in this hack, here are the calculations demonstrating the probability of winning:

Wager	Number of winning outcomes	Calculation	Resultingproportion
5 or 9 versus 7	8 versus 6	8/14	.571
1 or 6 showing	20	20/36	.556
2 or 12 versus 3	2 versus 2	2/4	.500
2, 3 or 4 versus 7	6 versus 6	6/12	.500
5, 6 or 7 versus 8 or higher	15 versus 15	15/30	.500

The "Wager" column presents the two competing outcomes (e.g., will a 5 or 9 come up before a 7?). The "Number of winning outcomes" column indicates number of different dice rolls that would result in either side of the wager (e.g., 8 chances of getting a 5 or 9 versus only 6 chances of getting a 7). The "Resulting proportion" column indicates your chances of winning.

You can win two different ways with these sorts of bets. If it is an even-money bet, you can wager less than your opponent and still make a profit in the long run. He won't know the odds are even. If chance favors you, though, consider offering your target a slightly better payoff, or pick the outcome that is likely to come up more often.