Category: Probability

When creating a blog, one of the first things to do is to come up with a name. Before choosing datastory.it, we considered several options, but some of them were already taken. On one of these sites, we came across a phrase that made our few remaining hairs stand on end. It went something like this:
“This site contains an algorithm capable of generating Lotto numbers that are more likely to be drawn than others.”

Such words sound to a statistician like a blasphemy sounds to a priest. Have you ever heard of “hot numbers” or “overdue numbers”? Surely you have. Well, we can guarantee you that these numbers are meaningless, and there is no algorithm capable of generating numbers more likely to be drawn than others. Let’s explore why.

The Lotto, or any similar game, consists of 90 balls, each with an equal probability of being drawn: 1/90, or about 1.1%. So far, so good. But what if you were told that after 100 draws, all the numbers from 1 to 90 had been drawn except one, say 27? Would it change your strategy? Would you bet on 27 because it’s overdue? The answer is an emphatic no because the probability remains the same for all numbers. There are at least two reasons for this.

Intuition – The balls in the drum are not influenced by previous draws. There is no reason why a ball should become larger, smaller, hotter, or colder depending on how many times it has been drawn (or not drawn) in the past. Each ball’s probability is always equal to that of the others, even if it hasn’t been drawn for a thousand consecutive rounds.

Probability Theory – This phenomenon can be described by a random variable following a geometric distribution, which measures the probability that the first success occurs after a certain number of trials (each with equal probability—in this case, 1/90). It can be shown that the overall probability remains exactly the same for both the n-th draw and the m-th draw, where m is greater than n. Thus, we arrive at the same conclusion: the drum has no memory.

The “Hot Numbers” Misconception – Where do these so-called Lotto experts go wrong with their theories on overdue numbers? They often invoke the nebulous dogma known as the Law of Large Numbers. While this is a complex topic that will be discussed separately on this blog, the law essentially states that as the number of trials approaches infinity, the frequency of each outcome converges to its theoretical probability—in this case, 1/90. For example, after 90 million draws, each number will have been drawn approximately 1 million times.

However, this law does not imply that after a certain number of draws, the probability of a number increases because it has been drawn less often than others. The reasoning of those who promote overdue numbers would only hold true if there were a finite number of draws. In such a case, if by the second-to-last draw one number had been drawn less frequently than the others, it would logically be the one drawn in the final round. But the Law of Large Numbers refers to an infinite number of trials, so with every draw, it’s as if everything resets. From that point on, the expected frequency of every number remains 1/90 for all future draws.

What we’ve discussed so far doesn’t just apply to the Lotto but extends to all situations involving independent repetitions of the same event, such as rolling dice or betting on numbers and colors in roulette. For example, if after 10 spins of a roulette wheel, red has come up every time, what do you expect on the next spin? If you think black is more likely, you’re deeply mistaken: the probability doesn’t change—it remains 50-50 for red and black (ignoring the green zero).

So, if you decide to play the Lotto or any similar game, don’t trust anyone urging you to bet on overdue numbers. Doing so would be no different than choosing numbers based on your birthdate or the last digits of your license plate.