Given the gambler's fallacy, why do we have a strong intuition of design if a remarkable result happens quickly in a game?

Question

In any sort of game that involves selecting one out of many opportunities where one of those opportunities is a win, and each trial is independent and random, each trial has an equal chance of winning. For example, throwing dice, no matter how many times it has been thrown before, each digit has an equal chance of 1/6. No matter how many times a coin has been tossed before, each side has a 1/2 chance of landing.

Suppose you are throwing dice. You're trying to get 5 sixes in a row. You've thrown these dice 400 straight times. On the 401st, 402nd, 403rd, 404th, and 405th throw, they land on 6. Five straight sixes! Finally, you think. It strikes you as impressive but after all, you'd been throwing for so long. This was bound to happen.

Now imagine another scenario. This time, you've thrown dice 400 straight times. You're tying to get five straight sixes and it just doesn't happen. You decide to try something new. You open a computer and use a random number generator that generates a random number between 1 and 6. You get a 6 the first five straight times. You're bewildered at your luck. It seems too good to be true.

Almost everyone, I would wager, would have an extremely strong intuition that something surprising has just occurred. Is it reasonable for you to assume, just based on the fact that you got five sixes in a row in the random number generator, that it's a wonky program? If so, why? The chances of getting five sixes in the second scenario, assuming the generator is random, is the same as the chances of the five dice rolls after the 400th try. After all, they're independent

Answer 1

Firstly, as someone who subscribes to the idea of determinism, I don't think anything is truly random. Now, you can say what you will about free will, but even if that truly exists, we can't program a truly random number generator. The last I read about it, the best random number generators were using the flickering intensity of light from distant stars as seed values. Of course those stars' lifespans, light emissions, etc etc were determined from the moment of their creation. The results appear random to us, but were really predetermined billions of years ago.

So, for the sake of argument, and to avoid the whole determinism scenario, let's just assume that we can generate truly random dice rolls and computers can generate truly random numbers.

In your proposed scenario, the only thing that has changed in the mind of the person rolling dice is the medium through which they are doing it. Humans are naturally predisposed to find meaning even if there is none to be found; this is well known and the foundation for many common logical fallacies. In this case it would be correlation vs causation. Yes, there is a correlation between switching from real dice to computer dice, and rolling 5 sixes in a row. But is this causation? If the computer is truly random, of course not. Humans, however, are notorious for confusing the two.

If you pay attention in your day to day life, you'll notice it all the time. For example, people will tell stories from their own life and generalize from personal experiences, to paint the whole world. "That person just ran a red light, drivers are idiots in this state." In reality, the location has little to no bearing on why the driver ran a red light, but if we can think of a reason for something to be the way it is, we will almost certainly assign that reason as an explanation even if it makes no logical sense.

So, to answer your question, it would be a case of confusing correlation with causation.

Answer 2

Probability, as the gambler's fallacy shows, can be counter-intuitive. Consider the case in which you roll the dice 400 times without success. You then hand them to your friend who rolls five straight sixes. Would that seem more surprising than rolling five sixes yourself after your 400 unsuccessful attempts? If so, then it is a delusion. The chances of any particular five consecutive throws being sixes are very small (one sixth to the power of five). But the chance of at least five consecutive throws out of a larger number being sixes is higher. You can generalise it to five consecutive events in any set where each event has a one in six probability, whether that be rolling a dice, picking a card from six cards, generating random numbers between 0 and six etc.

Answer 3

The idea of mathematically independent dice throws with probability exactly 1/6 is just a theory. You know it is not a complete accurate description of the real world because that is impossible but in your situations it is reasonably good approximation that allows one to make predictions about the real world that will be true often.

This is true in both your scenarios but if you think about them this way you can see where the difference comes in. You know you are only modelling reality and depending on how well the model did in the path you adjust your confidence in the model.

In the first case you have a pattern of 406 dice throws and so your model of mathematical independence fits the real observations quite well. So you see now reason to question the model.

In the second case the computer random number generator only generated 6 numbers and every single one of them was a 6. Based on these observation you doubt your model, it seems questionable whether you are actually seeing independent dice throws.

Note you could have the same effect by using a different die. If you observe the pattern you described you might suspect the second die is loaded.