The decision after the roll
In many dice games the dice are only half of the game. The other half is what you do with them: keep a result, reroll some of it, or press on and risk what you have. Those roll decisions can be worked out with a small amount of arithmetic, and the same few questions come up every time.
- What am I trying to achieve with this roll?
- What does each option give me on average?
- What do I lose if it goes badly?
Rerolling a single die
Suppose your goal is simply a high total, and you may reroll any die once. A fair d6 averages 3.5 (the mean of 1 to 6). If the die currently shows a 3, a reroll has an average result of 3.5, so rerolling is slightly better on average. If it shows a 4, rerolling averages lower than what you hold, so you keep it. The break-even point is the average of the fresh roll, and anything below that is a candidate for rerolling.
This rule only holds when you are maximizing a plain total and the reroll is free. If a reroll costs you something, or you are aiming at a specific target, the threshold moves.
Rerolling toward a target
When you need a specific result, the question becomes how many chances you have left. The chance of getting a particular face at least once in two rolls of a d6 is 1 - (5/6)^2 = 11/36, about 30.6%. With three rolls, it is 1 - (5/6)^3 = 91/216, about 42.1%. A reroll helps, but each extra attempt adds a little less than the one before, because you are only counting the cases that earlier tries missed.
For a combination of several dice, such as five dice needing to match, a useful habit is to hold the face you already have the most of and reroll the others. Getting all five of a kind in one throw has probability 6/7776, or 1/1296, so starting from a promising set matters. This is a general-purpose approach, not a guarantee: a different plan can be better if the scoring rewards other combinations, and in the end the dice may still refuse to cooperate.
Push-your-luck games: the break-even point
In a push-your-luck game, you keep rolling to build a turn total, but one bad result wipes it out. A common simple form: roll one d6. On a 1, you lose everything you have gathered this turn. On 2 to 6, you add the number to your total. You may stop at any time and bank the total.
Should you roll again when your turn total is T? Compare the average gain with the average loss.
- With probability 1/6 you lose T.
- With probability 5/6 you gain a number from 2 to 6. Those faces sum to 20, so across all six faces the expected gain is 20/6.
The expected change from one more roll is therefore 20/6 - T/6. This is zero when T = 20. Below 20, a further roll is favorable on average; above 20, the expected change turns negative. That is where the well-known “stop around 20” rule of thumb comes from for this type of game.
Be careful about what that means. It is the break-even point for a single additional roll, not a complete plan. When you are far behind in a longer game, a riskier approach may make sense, and when you are about to win, a more cautious one may. Even a break-even rule is a statement about averages: on any given turn, you can follow it perfectly and still lose the turn.
Expected value is not the whole story
The calculation above treats gains and losses as equal in worth, which is not always right. A game can reward stability or punish a single bad turn more than the average suggests. Two options with the same average can feel very different:
- Option A: always gain 3.
- Option B: gain 6 half the time and 0 half the time.
Both average 3. If you need at least 6 to avoid elimination, B may be the better choice despite the same average. If you only need to keep pace, A is safer. That is a calculated risk: you are choosing a variance level to match the situation, not just maximizing the mean. A neighbouring idea, risk management, covers how to think about losing small amounts in exchange for avoiding large ones.
Sunk effort and streaks
Two habits tend to distort roll decisions.
Sunk effort. If you have been building a large turn total, it is tempting to protect it by continuing, or to keep going because you have already invested so much. The dice don’t know how much you have gathered. What matters is what the next roll risks compared with what it gains. A large pot actually makes stopping more attractive, because more is at stake.
Streaks. After several good rolls, a player may think the dice are “hot,” and after several bad ones, that a good roll is due. Independent dice do neither. Each roll has the same chance as before. Strong emotional decisions, whether from frustration or excitement, often come from reading patterns into noise.
A simple routine for any roll
- Write down the possible results and how likely each is.
- Work out what each option gives you on average. Include the cost of failure.
- Ask whether the situation calls for steady results or for a higher-variance attempt.
- Take the option that fits, and accept that it may still go wrong.
This sort of reasoning supports informed decisions, but it never removes the chance element. Even the best choice can lose on a given roll. What the arithmetic can do is stop you from making the choices that are poor on average, and help you avoid paying for hope.
When long shots make sense
A long shot is not always a mistake. If you are very far behind, a low-probability plan that offers a real chance of catching up can be reasonable, because the safe plan almost surely loses. A 1-in-36 result, for example, is a rare event: about 2.8%. It might be a sensible gamble only when the alternative is close to hopeless, and the costs of trying are small. Treat it as a way to use the position you are in, not as something likely to work. The honest summary of long shots is that they come off rarely, and most of the time they don’t.