Is the case of repeated Bonoloto numbers so rare?

Image of the Bonoloto draw on Saturday, March 11.

On March 11, the winning combination in the Bonoloto turned out to be almost identical to the one that had been obtained just two days before, on March 9.

The numbers of the first draw were repeated: 08, 21, 23, 40, 43, 47... With the exception of 43, which became 28. The complementary number (26) and the refund (7) were also repeated.

This coincidence caused all the alarms to go off and accusations began about the possible manipulation of the draw.

But is it really so strange what has happened?

The bonoloto is a draw where the possible results are counted in the millions and the probability of hitting a six-figure combination is very small, much less than that of winning in the Christmas Lottery: one in 13,983,816 times or one probability of 0.0000000715 or 0.00000715%, depending on how we want to express it.

These figures give us the measure of uncertainty about the occurrence of an event.

If we use the expression between 0 and 1, 0 indicates that the event is impossible and 1 that the event will occur for sure.

The percentage shows the number of times the event will occur if the situation were to repeat itself 100 times.

This latter interpretation is known as frequentist.

In draws like the Bonoloto, each new repetition is completely independent of the rest.

This implies that the draw has no memory and that every time the numbers are extracted, a specific combination has the same probability of coming out.

Perhaps we would have been less surprised if instead of coming out the same numbers the consecutive ones of all of them had come out —09, 22, 24, 41, 44 and 48—;

or if a combination of numbers had come out where they were all even, or all were prime... This happens due to something called "the gambler's fallacy", a cognitive bias that makes us think that what just happened is more difficult for it to happen again .

The gambler's fallacy is a cognitive bias that makes us think that what just happened is more difficult for it to happen again.

It is also true that surely, when this result surprises us so much, it is because we are interpreting it in another way.

If we compare the probability of rolling that number —in general— with the probability of rolling any other number, obviously the latter wins.

Because any other number coming out accumulates the probability of all the values, except the one in question and that probability is very high.

Specifically, 1 minus the probability of a combination, which is practically 1. So, yes, seen like this, it was more likely that any other number would come up than the one that has come up.

But that, again, goes for any combination.

We might wonder how it could have happened so often in time.

According to the laws of probability, how many draws have to take place for us to see a number that has already come out again?

A very gross simplification of this would be to toss a coin and think about how many times you would have to toss it for it to come up heads again.

Taking into account the probability of a specific extraction, the average time that we must wait to observe a value, one in particular, whatever it is, is 13,983,816 draws.

Not surprising, considering that this is the number of possible combinations.

Any other number was more likely to come up, but that goes for any combination

Seems like a long time, doesn't it?

So it is strange that it has happened so often... Well, again, the answer is not so much.

Because when we give the mean of a variable, also known in probability as the expected value, our mind gets used to the idea that the repetition will occur every certain number of draws and only then.

In other words, we think that the number of repetitions is very likely around that number and values ​​far from that time interval are very unlikely.

However, this is not true in this case, since the time between repetitions can take any value with very low probability.

In fact, even if its mean is a specific number, low values ​​—that is, short times between repetitions— are somewhat more likely than higher ones, although, in all cases, the probability is very small.

In short, the event was rare, yes, but like any other event, any other combination and at any other time.

This, coupled with the idea that the simplest explanation is always the correct one, leads me to believe that rigging the draw is not the most likely possibility.

Anabel Forte

Ágata Timón García-Longoria

Coffee and Theorems

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