Jan Hvizdak

Karpatska 3

Kosice 04001

Slovakia

Let's take a look at lotteries such as Lotto, or in general any other game where one has to pick numbers.

From the statistical point of view, any combination of numbers carries the same weight and so the probability of the combination 1, 2, 3, 4, 5, 6 is the same like the probability of the combination 41, 42, 43, 44, 45, 46. However, lotteries aren't usually true random numbers generators. A true random numbers generator is completely unpredictable, but a lottery machine can be predictable under some circumstances, this assumption is based upon study of several lotteries including Keno in which particular combinations do not tend to be drawn as often as others even if the analysis consists of thousands of draws.

In order to simplify things, let's not consider combinations, but think about single numbers only. If there are say 50 numbers in the draw and if you let it run 50 times, each number should be present in one draw in average. Naturally, it doesn't work this way and for example the result will be :

number 0 – drawn 0 times |

number 1 – drawn 5 times |

number 2 – drawn 2 times |

and so on... |

Now, if you repeat draws 1 000 000 times, the aberrance should lead to zero and each number has to be drawn 1 000 000 / 50 times which is 20 000. However, this is just a theory and the actual numbers will vary. No matter that the results may vary for each number, the aberrance shouldn't be too high. In 1 000 000 draws, any number shouldn't be drawn more than 26% and less than 24%. If it's so, then the lottery isn't using a true random numbers generator for sure.

Generally if a number hasn't been drawn in the last 100 draws, and if there are only 50 numbers to choose from, the probability of being drawn raises as one could consider. In fact, the probability raises only in statistical point of view.

Practically a number doesn't have to be drawn even a one time during 1 000 000 draws, but a true random numbers generator or a pseudo-random numbers generator **MUST** generate the same result for each number in a long time interval. In this term, if a number hasn't been drawn for too long, the time when it's going to be drawn is close. Because the probability of being drawn the next time is still the same!

No-one can tell when a number is going to make a break-through and when it's break ends as long as the algorithm of our random numbers generator is not known. I.e. we've studied several lotteries including Keno10 in Slovakia as we come from this country and there are cases of combinations of 4 numbers not being drawn for over 2 000 draws even though their theoretical probability of being drawn would say that a break of 2 000+ days isn't even possible, or is very unlikely.

The same principle as above applies to sets of numbers; No matter if you're going to bet on 2 numbers or 6 ones. The break is just a variable that depends on how many numbers we have and how many are going to be drawn.

One number's average break is 4.

Two numbers' average break is roughly 17.

Three numbers' average break is approximately 72.

Ten numbers' average break is exactly 8 911 711.

As you can see above, one number's average break is only 4 while 10 numbers' average break is almost 10 million.

It's possible to bet on a single number with currently long break. In most games you'll win 2x if you pick one number correctly. If you don't pick the right number, you lose. So the example:

Your chance to pick 1 number correctly is 32.5% which is 1:3. 3 is also the average theoretical break for each number. One could think that it's not so likely to find a number with a break of 60 (say) days. If such a person decides to bet on a number that hasn't been drawn for the last 55 days, statistically it shouldn't be hard to win. On the other hand, betting each day $5 on one number won't bring you any profit if there are next 20 days without that number and it's finally drawn on 21th day. You'll lose 20*$5 ($100) and win only $5 ($10-$5). That's why it's necessary to increase the bet according to your total investments.

Day 1 | $5 |

Day 2 | $10 |

Day 3 | $20 |

Day 4 | $40 |

Day 5 | $80 |

Day 6 | $160 |

Day 7 | $320 |

Day 8 | $640 |

Day 9 | $1 280 |

Day 10 | $2 560 |

Day 21 | $5 242 880 |

There are two problems; Firstly, it's necessary to have plenty of money available for this scenario. No-one can guarantee that on 21^{th} day you're going to win. The break can last 30 or 40 days! In addition, your total earnings will be $5 anyway! I wouldn't risk over $500 000 for some $5!

This strategy can be effective if only you're aiming at more than 1 number and if the potential income is much higher than potential loss.