The Birth of Probability — Roll the Dice and Find the Odds

In 1654 a gambler asked Blaise Pascal how to split the stakes in an unfinished game. Pascal and Pierre de Fermat worked out the answer in a famous exchange of letters -- and invented the mathematics of chance. You will rediscover its first lesson with two dice.

Take two dice. On paper, make a tally chart with a row for each possible total from 2 to 12. Now roll the two dice fifty times, and each time add a tally mark next to the total you rolled. Take your time and be honest with every roll.

Turn your tallies into a little bar chart. You should find the middle totals -- especially 7 -- came up far more often than 2 or 12. Why? Because there are six ways to make 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) but only ONE way to make 2 (1+1) or 12 (6+6). More ways means more likely.

What your dice teach. (1) When outcomes are equally likely, a probability is just favourable outcomes divided by all outcomes -- 6 ways out of 36 for a seven, so one in six. (2) Your fifty rolls wobble around the true odds; a thousand rolls hug them tightly -- the 'law of large numbers'. (3) The counts of ways to make each total are hidden in Pascal's triangle, tying chance to counting. (4) The reasoning Pascal and Fermat invented for a gambling dispute now underlies insurance, weather forecasts, medical trials, quantum physics and the machine-learning behind modern AI. A question about dice became the mathematics of uncertainty itself.