1.1 Dice rolls and probability

Much has been won and lost over the years in games of chance, such as dice and cards. But how much chance do you actually have of winning? And can it be worked out mathematically? When you roll a die – assuming it’s fair – it’s equally likely to land on any of its six sides. And because there are six sides, your chance of throwing a six is one in six, or a sixth. Of course, that’s exactly the same as your chance of getting anything else, a one, two, three, four or five, because the die is fair. That may feel a bit odd because when playing games, we’re often hoping to throw a six, and it can feel like that doesn’t happen often. But of course, it happens just as often as getting any other number. Even if you roll the die lots of times and it lands on other numbers, that never makes it more likely that you’re going to throw a six the next time. The past throws don’t influence the current throw – you’re not ‘overdue’ for a six. The chance stays exactly the same every time you throw: 1/6.

In maths, we call the measurement of chance ‘probability’, and we often use the letter p for short. Probability measures the likelihood of an event occurring, and it’s expressed as a number between zero and one. An event which is certain, such as the die landing on any of the numbers from one to six, has a probability of one. It’s a certainty because there are no other possible outcomes. Well, not in normal circumstances anyway. So, if you add up the probabilities of the die landing on any of the six individual sides, the total comes to one. An event with an equal chance of occurring or not occurring, such as the die landing on an even number, has a probability of a half, because exactly half of the numbers on a die are even: two, four and six. An impossible event, such as the die landing on none of its six sides, has a probability of zero. Working out the probability tells us how likely it is that something will happen. To do that, you divide the total number of events – the things you’re interested in – by the total number of possible outcomes – the number of things that could happen.

So, the probability of rolling a five is 1/6, because there’s only one event you want – rolling a five – but there are six outcomes that could happen. Whereas if you’re playing a board game, and you need to roll a five or more to win, then you’re happy with a five or a six, which means you have two events you’d be happy with out of the six that could happen. So, the probability of you throwing the score you need to win that game is 2/6, or 1/3.

But what if you have two dice and you roll them together? What is the chance of getting any particular outcome then? There are six possible outcomes for each die. Whatever number comes up on the first die, there are then six possible outcomes for the second die. So, there are six times six, or 36 possible outcomes in total. So, for example, the first die came up with a three, so the second die has six different numbers it could combine that with. It could be three and one, three and two, three and three, three and four, three and five, or three and six. These are all the possible outcomes with two dice, where the number in red represents the outcome for the first die, and the number in blue represents the outcome for the second die. The number order is important here. Throwing a two followed by a three is different from the event of a three followed by a two. But what if we’re interested in the combined result – the total number produced by the dice – rather than the results for the individual dice? In this case, providing the two numbers add up to the same total, neither the order nor the actual numbers on the die matter. Some results can be obtained in multiple ways. For example, a total of five can be obtained in four possible ways: I could have a two and a three, or a three and a two, a four and a one, or a one and a four. Other results are less common. For example, a total of 12, that can only be obtained by rolling a six and a six. In maths, we would say that the frequency (often shortened to f) of getting a particular score is the number of ways that can happen. So, the frequency of scoring five with two dice is four, because there were four ways that I could roll the dice with a total of five. And the frequency of scoring 12? Well, that’s one, because there’s only one way. In each case, the number of possible outcomes is 36, but the event frequency is different. So, the probability of getting a total of five cannot equal the probability of getting a total of 12. The probability of scoring five is 4/36, or 1/9. And the probability of scoring 12 is only 1/36. So, if you roll two dice, you’re more likely to score five than 12. Of course, if you roll a pair of dice 36 times, you’re not likely to get exactly the scores that the theoretical probabilities may seem to predict. But, if you roll a pair of unbiased dice many, many times, then your results will get closer and closer to the event frequences we’ve just worked out mathematically. In short, real life is not completely predictable.

[IN THE STYLE OF A QUICKLY SPOKEN LEGAL DISCLAIMER] Caution! Maths cannot be held responsible for success or failure in games of chance. Your house is at risk if you do not keep up repayments … or stake it on a game of dice. Players should be over 18 to gamble … and anyone with a sound understanding of probability will probably realise that the house always wins!