Probability with a fact attached
Ordinary probability answers questions like “what is the chance this card is an ace?” Conditional probability answers a narrower question: “what is the chance this card is an ace, given that I already know something?” The “given” part shrinks the set of possibilities, and the probability is recalculated within what is left.
The standard notation is P(A | B), read as “the probability of A given B”. It is worth learning to read it, but the idea can be handled entirely by counting.
A revealed card
Start with a standard 52-card deck containing four aces. Before anything is dealt, the chance the top card is an ace is 4 out of 52, which is 1 in 13, or about 7.7 percent.
Now suppose the top card is dealt face up and it is an ace. What is the chance the next card is also an ace? Three aces remain among 51 cards, so the chance is 3 out of 51, or 1 in 17, which is about 5.9 percent. The deck has changed, and so have the odds.
Compare that to the chance of drawing two aces in a row from a fresh deck, before either card is seen. That is 4/52 × 3/51 = 12/2,652, which simplifies to 1 in 221, or about 0.45 percent. The two numbers answer different questions. One is “how likely is the second ace once the first is known?” and the other is “how likely is the pair, before any card is known?”
The counting method
You can always solve these by listing the cases that remain.
Say a friend draws a card from a deck and tells you only that it is a face card (jack, queen or king). There are 12 face cards, and 4 of them are kings. The chance the card is a king, given it is a face card, is 4 out of 12, or 1 in 3.
Or take two dice. Without any information, the chance that the two dice total 8 is 5 out of 36, about 13.9 percent, because five of the 36 equally likely pairs add to 8. Now suppose you learn that the first die shows a 3. Only six pairs remain possible, and only one of them, 3 and 5, totals 8. So the chance becomes 1 in 6, about 16.7 percent. A single known fact changed the answer.
Now suppose the first die is a 6. A total of 10 or more needs the second die to be 4, 5 or 6: three of six possibilities, so 1 in 2.
The formula, when you need it
In symbols: P(A | B) = P(A and B) ÷ P(B).
Check it with the face-card example. The chance of being a king and a face card is the chance of being a king, which is 4/52, since every king is a face card. The chance of being a face card is 12/52. Dividing: (4/52) ÷ (12/52) = 4/12 = 1/3, matching the counting result.
The formula is just the counting method with probabilities in place of counts: look at the part of the world where B is true, and ask how much of it also has A.
Watch the direction
A frequent mistake is treating P(A | B) as if it were P(B | A). They are generally different.
Using the same deck: the chance that a card is a face card, given that it is a king, is 1, because every king is a face card. But the chance that a card is a king, given that it is a face card, is only 1 in 3. Reversing the condition changes the answer completely.
This reversal error is behind many misleading claims. “Most winners used this approach” is a statement about winners; it says nothing about how often the approach leads to winning. The tool for converting one direction into the other is Bayes theorem, which combines the reversed probability with how common each outcome was to begin with.
Dependent and independent events
Whether new information changes the odds depends on whether the events are connected. When one event does not influence the other, learning about it changes nothing: after a fair die shows a six, the next roll is still 1 in 6. Cards drawn without replacement are different, because each card removed changes what is left, which is the idea behind dependent events.
This is also why counting outs in a card game involves recalculating after every revealed card, and why a deck is described as having memory while dice and coins do not. A flipped coin that has landed heads five times is still 50-50 on the next flip. The past tells you nothing there, because the flips are independent.
A note on hidden information
In games with hidden cards, you often condition on things you can see: your own hand, cards on the table, discards. Each visible card removes possibilities, and your estimate for what an opponent might hold should reflect that. The careful part is deciding what you actually know, as opposed to what you suspect. Conditioning on a guess presented as a fact gives a precise-looking number with little behind it.
Conditional probability helps you understand the structure of a game. It does not give you an edge against rules that are set to favour the house, and outcomes remain uncertain. If gambling is becoming a problem for you, a local support service can help.