| (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
| (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
| (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
| (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |
| (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |
| (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |
If you look at the long diagonal starting in the bottom left, you can see that there are six ways out of a possible thirty-six to reach a total of exactly 7.
So for two six-sided dice the probability of rolling a seven is
P(7) = 6/36 = 0.1667
Or about one in every six tries.
That question was too easy, so let's make it a little bit tougher. What's the probability of rolling seven 7s in a row? Okay, it's not that hard if you know how to calculate the probability of independent events. You just need to multiply the probabilities of each event. In this case we want to multiply 0.1667 by itself seven times,
0.1667 * 0.1667 * 0.1667 * 0.1667 * 0.1667 * 0.1667 * 0.1667
or we can just raise it to the 7th power.
(0.1667)7 = 0.0000036
You can see that rolling seven 7s in a row has some pretty long odds. That will only happen about 36 times in 10 million tries!
But what if you've already rolled six 7s in a row? What's the probability of rolling a seventh 7? Before we do any calculations, would you say that the odds are pretty good or pretty bad of getting a seventh 7?
If you think the probability is extremely bad, you're falling for a very common cognitive bias called the Gambler's fallacy. This fallacy accounts for the belief that a streak of good or bad luck is "due" to change. A gambler who has had a long streak of losing hands in blackjack might decide to increase his bet, banking on the feeling that his luck is about to change.
The Gambler's fallacy arises from out intuitive sense that things have a way of "evening out" in the long run. The truth is that over the very long run, things do even out. This is called the Law of large numbers. If I flip a coin a million times, I can expect to get reasonably close to half-a-million heads. But if I flip the same coin only ten times and get ten heads in a row, the probability of seeing a head on the next flip is still exactly one-half. The Law of large numbers only applies to the average of a large sample, not to individual coin flips, hands of blackjack, or rolls of the dice. The coin (or the cards, or the dice, or the universe for that matter) doesn't remember the result of the previous trials, and therefore cannot be influenced by them.
Oh, I almost forgot, the probability of getting that seventh 7 after rolling six in a row is 0.1667, exactly the same odds as getting a 7 on any other roll. The dice have no memory.
