What a mulligan decision really compares
A mulligan lets you discard a poor starting hand and draw a new one, usually with a cost. In many games the cost is having fewer cards. The decision therefore compares two things: the hand you hold, and the average of what you would get by starting over with a smaller one.
Neither side of that comparison is certain. You are not asking “is this hand good?” but “is this hand better than the typical hand I would get if I redraw, after accounting for the lost card?” Rules vary a great deal between games, so check the exact procedure in yours. The reasoning below applies to any version where a redraw costs you something.
Information you already have
The key observation is that your hand tells you something about the rest of the deck. If you hold seven cards, the remaining deck is smaller and its makeup is known. That is conditional probability at work: your chances of drawing what you need next depend on what you have already seen.
Take a 40-card deck with 17 resource cards. You open with seven cards containing exactly one resource card. The remaining 33 cards contain 16 resource cards and 17 other cards.
- The chance of drawing at least one resource card in your next two draws is one minus the chance of drawing none: 1 minus (136 divided by 528), which is about 74 percent. The 136 is the number of ways to choose two of the 17 non-resource cards, and the 528 is the number of ways to choose any two of the 33 cards.
- In three draws it is 1 minus (680 divided by 5,456), about 87.5 percent.
So a one-resource hand is far from hopeless. It also means that in roughly one case in four, you reach your second turn still stuck at a single resource card, and one in eight you are still short after three draws. Whether that risk is acceptable depends on how cheap your cards are and how much the format punishes a slow start.
How often do you start with a normal hand?
For the same 40-card deck, a seven-card hand contains two to four resource cards about 79.5 percent of the time. That means about one hand in five falls outside that band: roughly 10.5 percent have zero or one resource card, and about 10 percent have five or more.
This is useful context. If you redraw a seven-card hand at random, you are choosing a draw that is workable roughly four times in five, but only after paying the cost. With a six-card hand from this deck, the chance of two to four resource cards is about 78.4 percent, which is barely lower. The resource count itself is little affected by the smaller hand; the real cost is one fewer spell to work with.
A simple framework
When deciding, it helps to ask these questions in order:
- Can I cast spells? Count the resources against the cheapest plays in the hand.
- Can I keep casting? Estimate the chance of finding more resources in the first few draws, as in the example above.
- Does the hand do something? A hand of resources and nothing else, or spells with no way to cast them, fails in different ways.
- What is the alternative worth? A six-card hand is typically a bit weaker than a seven-card one, so a marginal seven should often be kept, while a seriously flawed one should not.
- What is my opponent likely to do? Against a fast opponent, stumbling early is costlier. Against a slow one, you can afford to wait for draws.
None of these steps gives a numerical answer on its own. They structure the judgement.
Extremes are easier than the middle
Hands at the edges are easy to evaluate. A hand with no resource cards in a deck that needs them almost always fails, and a hand with every card a resource has no plan. The hard decisions are the borderline hands: two resource cards and expensive spells, or one resource card with several cheap spells.
For borderline hands, the numbers above help. A hand with one resource card and several cheap spells that can be played with the resources you are likely to draw is much more acceptable than one with the same resource count and nothing castable. The chance of finding the next resource card is the same, but the cost of waiting is different.
Playing first or second
Many games give the player who goes second an extra card. An extra draw increases the chance of finding the resource you are missing, which can justify keeping a slightly weaker hand when you will see more cards before acting. How much it helps depends on the game, so check the turn structure before applying this idea.
Why you cannot judge a mulligan by one result
Imagine you keep a risky hand and win, or redraw and lose. It is tempting to conclude that the decision was right or wrong. A single game says very little. A choice with a 75 percent chance of working fails one time in four, and that failure does not prove the choice was bad. Judging decisions by their outcome is a common mistake, related to the broader pattern of overconfidence in assessing odds.
A better habit is to evaluate the decision with the information you had at the time, and to look at results over a large number of games. Even then, ordinary short-run results can distort your view, so do not adjust your rule after a few unlucky hands.
Limits of the maths
- Each calculation depends on your deck list. A different resource count or curve changes the figures.
- The estimates assume a well-shuffled deck.
- The framework assumes you can judge how good the spells in your hand are. In practice that judgement is the harder part.
- Some games allow you to choose which cards to put back after a redraw, which changes the arithmetic. Use the rules of your game.
A fair summary is that keeping a flawed hand is a risk with a calculable chance of working, and redrawing is a different risk with its own cost. The numbers let you compare them; they do not remove the uncertainty.