# how lotto probabilities work

More specifically, what’s the term for 1 2 3 4 5 6 seeming less probable than any other 6 numbers between 1-75, yet still technically having the same odds? Are patterns taken into account when determining probabilities? Could I eliminate a few patterns (consecutive numbers, every other number, fibonacci, etc) and make a significant dent in the shear probability of winning?

In: Mathematics

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All numbers have the same probability of occurring. The only reason patterns and sequences of numbers *seem* less likely is because most combinations of 7 numbers don’t follow a distinguishable pattern or sequence, so when they do it seems novel and rare. This doesn’t mean they’re any more or less likely to occur though.

The term is “gambler’s fallacy”. It also applies when you roll a dozen 10s in a row, you think there’s no way you can roll another one, but the dice don’t remember what the last 12 rolls were, and the likelihood of rolling a 10 is still 1 in 12.

As far as how exponentially quickly the odds are of matching 6 numbers, it’s something like this:

Let’s say you’re matching six numbers from 1 to 35. The odds of getting all of them—from one batch of numbers with no repeating numbers—is 1 in (35x34x33x32x31x30) which is 1 in almost 1.17 BILLION.

That’s why most of those games use either less numbers in the field (25 numbers as opposed to 35 or more) or you only have to match 4 or 5 for the jackpot. This is closer to how New York’s Take 5 game works. It’s a nightly drawing with 5 numbers being drawn from 1-39. The odds of winning the jackpot (usually somewhere around $40,000-60,000–it’s a pari-mutuel game which means that the prizes are determined by how many people play and how many people win what prizes, ensuring the game is always profitable for the lottery) the odds are 1 in (39x38x37x36x35) which makes it 1 in a little over 69 million (nice).

It’s very easy to see how adding that sixth number drives up the odds so much, even from a smaller field of numbers. And the reason why it’s 39x38x37, etc. is because the second number has to factor in that one number has already been drawn.

It might be better to think of the numbers on the lotto balls not as numbers but as weird symbols without further meaning, it shouldn’t matter to the matter at hand to replace the numbers on the balls with different triangles, or make every ball have a different color, right?. The main thing is that you can recognize each and every individual ball, nothing else. For the future, let’s stick with colors.

This way, you look at a set of 75 colors and you draw 6 out of these colors. As you draw completely randomly, every set of colors has the same probability to occur. The amount of 6-element subsets of a 75-element set is written as binom(75, 6), since every set has equal probability to be drawn, we get 1/binom(75, 6) as our probability of a single one of them occuring.

Since we can express this problem without having patterns, the **patterns do not matter at all.**