What an out is
An out is a card that would improve your position in the way you are hoping for. In rummy-style games it is a card that completes a set or run. In bridge it can be a specific card in a specific hand that makes a contract succeed. The method is the same in each case: count the cards that help, count the cards you cannot see, and compare the two.
The basic formula is simple. If there are o helpful cards among u unseen cards, the chance that the next card you receive is one of them is o divided by u. Everything else in the topic is care about what counts as helpful and what counts as unseen.
Unseen cards, not cards in the deck
The number to divide by is the number of cards you cannot identify, which is not always the number left in the stock. Cards in your own hand are known, and so are any cards visible on a discard pile. Cards held by opponents are unseen too. From your point of view, a card in the stock and a card in an opponent’s hand are equally unknown, so both belong in the count, at least until you learn something about their hands.
Take gin rummy as an illustration. You hold 10 cards, and one card is face up on the discard pile. That makes 11 known cards and 41 unseen out of 52 (52 minus 11). If you need one of 2 specific cards, the chance that your next draw from the stock is one of them is 2/41, about 4.9%. Strictly speaking, that is the chance for the first card of the stock when you know nothing about the opponent’s hand. Opponent behaviour, such as picking up a discard, can shift the real figure.
Rummy examples
Suppose your hand includes the 7 and 8 of hearts. A run needs the 6 or the 9 of hearts, so you have 2 outs. If the 6 of hearts has already been discarded, it is a dead card and you are down to 1 out, the 9. Dead cards matter more than people expect, because they shrink the numerator without changing the size of the pile much.
Now suppose you also hold a pair, the 5 of spades and the 5 of hearts. A set of three fives needs one of the other two fives, the 5 of diamonds or the 5 of clubs. That is a different draw with 2 more outs. Because these cards all complete something, you can add them: 2 + 2 = 4 outs, and the chance your next card helps one of them is 4/41, about 9.8%.
Adding outs is only safe when the sets of cards do not overlap. If a single card could complete two different melds, count it once, since it is one card in the pile. Double counting is the most common arithmetic mistake in this topic, and it connects to the inclusion-exclusion principle, which exists to correct exactly that.
Rummy also shows the difference between “improves my hand” and “wins”. Some outs only reduce your leftover cards, and others finish the hand. Decide which you are counting before you count.
Looking more than one card ahead
The figure above covers one draw. Over two draws, the chance of hitting at least once is easier to find by looking at the chance of missing both. With 4 outs among 41 unseen cards, the chance of missing on the first draw is 37/41, and of missing on the second, given a first miss, is 36/40. Multiplying gives (37 x 36) / (41 x 40) = 1332 / 1640, about 81.2%. So the chance of hitting at least once is about 18.8%. This is the complement counting approach, and the second factor shrinks because the pile has fewer cards, which is the effect of drawing without replacement. In real play, though, the opponent’s discards and draws keep changing which cards are unseen, so a two-draw figure is a rough guide.
The poker comparison
Poker players use the same idea, and counting outs there often leads into quick approximations such as the rule of four, plus questions of pot odds. Those belong to poker-specific articles. The point for this article is that the underlying step, outs divided by unseen cards, is shared by every game in which a card you do not hold can finish a hand.
Bridge
Bridge has no drawing, but it has the same counting. During play, a declarer works out which cards are outstanding and where they might be. If the queen of a suit is missing and nothing is known about the opponents’ hands, the queen is in one of the two opposing hands, and before other information it is equally likely to be in either. With no other clues, a two-way guess like this is a 50% proposition. Players try to improve on that using extra information, such as how the opponents bid or which cards they have already played.
Counting cards played is the bridge counterpart of tracking dead cards. As tricks go by, the unseen pile shrinks, and the possible layouts of the remaining cards shrink with it. By the time a few tricks are left, the count of outstanding cards in a suit can leave only one or two layouts possible. This is also where the earlier discussion of hand distributions gets used: the starting split figures are only a starting point.
Common mistakes
- Counting cards you hold or have seen as outs.
- Forgetting that a discarded or exposed card is dead.
- Adding outs that overlap, so a single card is counted twice.
- Treating the stock size as the unseen count when opponents’ hands are also unknown.
- Assuming that a good chance of hitting an out guarantees anything. A 10% chance fails nine times in ten, and an 80% chance still fails one time in five.
Counting outs is bookkeeping for uncertainty. It tells you how likely a draw is, not whether it will come in, and it gives no edge in any game where the structure of the game costs the player money over time.