Random experiments to explore outcomes

We have all met simple examples of random experiments in common board games, played by repeatedly rolling a dice, flipping a coin, spinning a roulette wheel, and so on.

For instance, a dice is a small cube with point spots on its sides, from 1 to 6.

It is used in popular board games to produce a random number from 1 to 6 that will determine the next move. In some games, two or even three dice are rolled simultaneously.

In such experiments, at each turn, the possible outcomes are known, but it is uncertain which one will occur – it cannot be predicted.

That’s exactly the purpose of randomisation: it helps to make the game ‘fair’, so that any one of the possible outcomes is equally likely to occur.

But if the same experiment is repeated a few times, you may expect some random patterns to occur, while some other patterns may look unlikely. For instance, the value 6 has a one-out-of-six chance to be observed in a single dice roll. To put it differently, a 6 appears, on average, once every six rolls.

Does this mean that if a 6 has not appeared five times in a row, it should necessarily turn up next? Of course not – the dice has no memory, so after five no-6 values in a row, the chances of seeing a 6 in the next roll stay the same: one out of six.

However, suppose you have rolled the dice 100 times. Because of symmetry among the possible values 1 to 6, you would expect that, roughly, the proportion of each value would be about (small 1/6) of 100, that is,  (small 100/6 = 16.667), so about 17 times.

Imagine now that the value 6 has appeared 23 times, more frequently than each of the other five values, 1 to 5. Furthermore, suppose that at one point a 6 turns up three times in a row, resulting in you winning the game. Your opponent then complains that the dice favours the value 6, so it is not a fair dice.

This imaginary scenario leads to the following questions:

• If a fair dice is rolled 100 times, how many observed 6s do you expect?
• Would it be unusual for a 6 to be rolled 23 times in 100 rolls?
• Would it be surprising to roll three 6s in a row at some point?