Jan Hvizdak

Karpatska 3

Kosice 04001

Slovakia

As mentioned in the article Statistics and Lotteries, there is always some time period that can be characterised as good for some numbers and bad for others. In order to see horse racing or football matches as a mathematical problem we can take each racer or team as a number in the lottery. It is necessary to be 100% sure that there is no corruption or manipulation, which is hard in general.

If we have a football league that consists of 20 teams, we can assign a number to each team. It would look like this:

Team A – 1 | Team B – 2 | Team C – 3 | ... | Team T – 20 |

Say that 1 week all teams will play 1 match, so that's 10 matches per week. It could look like there are 20 balls in the lottery, and 10 are going to be drawn. Unfortunately, some matches can end 1:1, 2:2 or similarly. Of course, each result carries the same probability in terms of statistics. In my opinion the result that carries highest probability is 0:0 as every match starts with this score. Let's get back to the matches and how they can end...

1 vs. 2 – possible results are 1, x, 2. In words: 1 match, 2 teams, 3 possible results. If there are 10 matches, 20 teams, there are 30 possible results which means our mathematical chance to tell right the result of 1 match is 1:3.

Now imagine that we want to be right when guessing results of 2 matches. These results are possible if Team 1 plays against Team 2, and Team 3 plays against Team 4:

Team 1 | wins, | Team 2 | wins. |

Team 1 | wins, | Team 2 | loses. |

Team 1 | wins, | Team 2 | draws. |

Team 1 | draws, | Team 2 | wins. |

Team 1 | draws, | Team 2 | loses. |

Team 1 | draws, | Team 2 | draws. |

Team 1 | loses, | Team 2 | wins. |

Team 1 | loses, | Team 2 | loses. |

Team 1 | loses, | Team 2 | draws. |

You don't have to be a rocket scientist to know this. It's 1:9. If we would like to tell results of 3 matches, it would be 0.3*0.3*0.3. And so on.

Number of matches | Probability of right bet | 1:x |

1 | 0.33333333333 | 3 |

2 | 0.11111111111 | 9 |

3 | 0.03703703704 | 27 |

4 | 0.01234567901 | 81 |

5 | 0.00411522634 | 243 |

6 | 0.00137174211 | 729 |

7 | 0.00045724737 | 2 187 |

8 | 0.00015241579 | 6 561 |

9 | 5.080526E-5 | 19 683 |

10 | 1.693509E-5 | 59 049 |

11 | 5.64503E-6 | 177 147 |

12 | 1.88168E-6 | 531 441 |

13 | 6.2723E-7 | 1 594 323 |

14 | 2.0908E-7 | 4 782 969 |

15 | 6.969E-8 | 14 348 907 |

16 | 2.323E-8 | 43 046 721 |

17 | 7.74E-9 | 129 140 163 |

18 | 2.58E-9 | 387 420 489 |

19 | 8.6E-10 | 1 162 261 467 |

20 | 2.9E-10 | 3 486 784 401 |

As you can see, telling the right results of 10 matches is practically impossible in terms of these statistics. However, we all know that some teams play better than others and that some do win more than rest. Additionally we know something about players, weather, conditions, rankings. If some team is supposed to win the whole league, we can say that it has to win 6 or 7 matches out of 10. Then the probability isn't 0.3, but 0.6 or 0.7 . If we have 2 such teams playing 2 different matches (the league doesn't have to be the same), then the probability of telling results of 2 matches is quite different. Say that 10 really good teams play against 10 really bad teams, so the probability is 0.7 for each. Then the above-shown table would look like this:

Number of matches | Probability of right bet | 1:x |

1 | 0.7 | 1.43 |

2 | 0.49 | 2.04 |

3 | 0.343 | 2.92 |

4 | 0.2401 | 4.16 |

5 | 0.16807 | 5.95 |

6 | 0.117649 | 8.5 |

7 | 0.0823543 | 12.14 |

8 | 0.05764801 | 17.35 |

9 | 0.040353607 | 24.78 |

10 | 0.0282475249 | 35.4 |

11 | 0.01977326743 | 50.57 |

12 | 0.0138412872 | 72.25 |

13 | 0.00968890104 | 103.21 |

14 | 0.00678223073 | 147.44 |

15 | 0.00474756151 | 210.63 |

16 | 0.00332329306 | 300.91 |

17 | 0.00232630514 | 429.87 |

18 | 0.0016284136 | 614.09 |

19 | 0.00113988952 | 877.28 |

20 | 0.00079792266 | 1 253.25 |

As you can see now, being right in 10 matches is not that impossible! The chance is now 1:35.4 . Basically you can simply calculate probabilities and depending on potential winnings, it's very easy to tell if a bet is worth it or if it's not.

As above, any such principle can be turned into mathematical problem and can be described as a set of probabilities. Lotteries too have these periods in which some numbers are drawn more often than others. But the main advantage of football matches, NHL league or tennis is that in all sports the probabilities are not as in random processes. If a team that leads the league plays against team that's last, it's very likely to see the better team win.

In lotteries (and random processes as a whole) it's very hard to determine the right time and say “this is the right time to bet on the number 22!”