Humans, with a few exceptions, find it difficult to envisage big numbers. Sure, someone like Robert Graham (the conceiver of Graham’s Number), or John Nash (he of A Beautiful Mind fame) might be able to deftly calculate large equations or identify massive numbers, but picturing those numbers is another matter; the human mind, even a beautiful one, is not equipped to do it.
It’s easy enough for us to picture what a hundred looks like, perhaps legs on a centipede or people on a double-decker bus. A bigger number, like 50,000, is a bit more difficult, but we can conceive of a packed football stadium; twenty football stadiums, then, for a million people. Grains in a standard bag of sugar?
Seventeen million approximately for a kilo bag. But it is at this point where we begin to think of the millions of sugar grains as one entity rather than millions of different units. A trillion? A googol (Not the search engine, but 10 to the power of 100, or one followed by a hundred zeroes)? A googolplex (10 to the power of a googol, or one followed by a googol zeroes)? Forget about it.
Similarly, we flap when it comes to probabilities. If we took a national lottery, where the winning six numbers, say from 60 balls, were 3, 12, 29, 30, 42, and 55 last week. We know that math tells us that the numbers 3, 12, 29, 30, 42 and 55 are just as likely be the numbers drawn this week as any other set.
But when pushed, most of us would react by saying “it won’t happen, though”. Similarly, it’s hard to conceive that numbers 1, 2, 3, 4, 5 and 6 would be the winning numbers ahead of a random combination like 4, 12, 18, 29, 41, and 50.
We are conditioned to think the past influences the present
This touches upon the concept of the Gambler’s Fallacy. In a nutshell, we are talking about the belief that an event is more likely to occur after an independent event occurred in the past.
If the number 1 was drawn in the lottery draw for seven consecutive drawings, then we are conditioned to think that it is less likely to happen in the next draw; although, some people might think the opposite with the concept of hot and cold numbers, but that’s another aspect of the fallacy.
But when we break it down, we know that the drawing of one number in the past should have no bearing on the next one. We know this to be true when we think about, but we don’t accept it.
Consider the turning of a card in a game of red dog poker. As you can see here with this version, www.casino.com/ca/table-games/red-dog/, it is a simple three-card variant of poker. When played with a single deck, the probabilities are reasonably easy to work out; as easy as blackjack.
But professional poker players warn about the gambler’s fallacy entering our psychology with the perception that we “are due” a certain outcome on the particular turning of a card. Three aces are just as likely (or unlikely) to appear in the next hand when they have already been drawn in the previous one.
Perhaps this is easier understood with another casino game – roulette – and the simple bet of red or black (leaving aside the green zero pocket). Online roulette games will display the previous results for players to use as a guide.
Why? In a sense, it’s useless information. If black came up ten consecutive times, the player believes the red number is “due”. It will be red’s turn, so to speak. But the math tells us that it is just as likely to be black again as it is red.
Chances of red or black are always the same
But here’s the kicker: The odds of getting ten black results in a row in roulette are 1 in 1,376 (on a standard European roulette wheel). That’s unlikely, for sure, but it does happen on a rare occasion. Reportedly, the record for repeat outcomes like this was 32 consecutive red results (the odds were calculated around 11 million to one) at the American Casino in 1943.
More pertinently, however, it’s the marrying of those overall odds and the independent event of the next wager where the human mind falls down. We find it so difficult to separate the 50/50 chance of the eleventh event from the ten previous outcomes. Even if we understand the math, we tend to dismiss it.
The irony, of course, is that understanding the gambler’s fallacy does not really help us at all beyond influencing our decision whether to gamble or not. Simply knowing that past events do not influence the outcome doesn’t lend us any insight into the next result.
Some scientists claim to have mastered roulette by using chaos theory, but most of us must leave it to chance; each event independent of the last.