### Taking Chances Extended

This article is mainly about probability , the mathematical way of looking at random chance. The ideas and applications of probability are widespread: it is the foundation of the science of statistics, it influences decision-making in business (and even in sport), it helps us understand the way diseases spread and can be controlled, and the patterns of catastrophes such as earthquakes or transport accidents. Plainly, it applies to studies of Lotteries, and other games involving coins, cards or dice.

I shall use games of chance to illustrate ideas of probability, but I am NOT suggesting that you play any particular game. Indeed, the more you know about probability, the better you appreciate the odds against you in “commercial” games. I have never bought a ticket in the UK National Lottery, I belong to no casino, my bets on horses are small and rare.

Even before Roman times, soldiers used bone dice in various games to help pass the time on long campaigns, but the first systematic studies of how chance works were made in the sixteenth and seventeenth centuries. For example, gamblers wanted to know the chances for the different totals when three dice are thrown. Pascal and Fermat considered the following problem (“the problem of points”):

Total | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

Chance | 1/36 | 2/36 | 3/36 | 4/36 | 5/36 | 6/36 | 5/36 | 4/36 | 3/36 | 2/36 | 1/36 |

#### Averages

#### Games

#### Coin matching games

#### Randomising your choices

#### How to win at the Coin Matching game

#### “Solving” this sort of game

#### A word of warning

#### Benford’s Law

- A World Atlas might list the countries in alphabetical order, and give, for each country, its area (in sq km, say), its population, and gross domestic product.
- A supermarket uses the bar codes to print off a list of how many of each item have been sold each week.

- Car numbers are allocated in sequence.
- All (or most) telephone numbers in a certain area of town may begin with the same first digit.
- Seats in a theatre, cinema, soccer ground will be labelled A1, A2, A3, etc., in order.

- Computers and some pocket calculators are programmed to produce sequences of numbers such 0.74520778, 0.225189362, 0.01839284, …, intended to be evenly scattered between 0 and 1.

Brighton | 146 134 |

Brinkworth | 1 139 |

Bristol | 387 977 |

Brixton | 1 030 |

Broadstairs | 21 670 |

Broadway | 2 503 |

etc. |

Initial Digit | Frequency |

1 | 32 |

2 | 19 |

3 | 13 |

4 | 9 |

5 | 9 |

6 | 9 |

7 | 4 |

8 | 1 |

9 | 4 |

TOTAL | 100 |

Initial Digit | Values using 100 x LOG( ) |

1 | 30.1 |

2 | 17.6 |

3 | 12.5 |

4 | 9.7 |

5 | 7.9 |

6 | 6.7 |

7 | 5.8 |

8 | 5.1 |

9 | 4.6 |

TOTAL | 100 |

Initial Digit | Frequency | Total |

1 | 32 | 46.44 |

2 | 19 | 44.31 |

3 | 13 | 46.95 |

4 | 9 | 39.62 |

5 | 9 | 48.62 |

6 | 9 | 58.35 |

7 | 4 | 29.68 |

8 | 1 | 8.94 |

9 | 4 | 36.86 |