Conditional Probability After One Card
How the chance of the next card changes once you have seen one card, with the ace example worked both ways.
Conditional probability is the chance of one event given that another event has already happened. The given part is not decoration. It changes the deck you are allowed to count. One exposed card is enough to show the change, and enough to show when people apply the wrong update.
The definition on a small scale
Write P(A given B) for the chance of A once you know B occurred. The formula is the chance of both, divided by the chance of B. You do not need the symbols to use the idea. You need a revised list of the outcomes that are still possible, with the ones ruled out by B removed, and you need those remaining outcomes still equally likely.
Start with a full deck. The chance the top card is an ace is 4/52, which is 1/13. That is an unconditional chance. Nobody has shown you anything. The moment a card is shown, you are in a different problem, and the 4/52 sentence is finished.
Given that the first card was an ace
Suppose the first card is turned and it is an ace. Three aces remain, and 51 cards remain. The chance the second card is an ace is 3/51, which is 1/17. You can also see the same result as a shrink from 4/52 to 3/51. Both the numerator and the denominator moved. Forgetting to shrink the denominator is how people keep saying 3/52 after a card has left.
The chance both of the first two cards are aces, before either is seen, is (4/52) times (3/51). That product is 12/2,652, which simplifies to 1/221. Notice that 1/221 is not 1/17. The 1/17 is the second-card chance after an ace was seen. The 1/221 is the chance of the two-ace sequence from the start. They answer different questions. Mixing them is the same family of error as the ones in common card probability mistakes.
Given that the first card was not an ace
Now suppose the first card is a king. Four aces remain among 51 cards. The chance the second card is an ace is 4/51, a little higher than the original 4/52. Information that a non-ace left the deck raises the ace chance slightly. Information that an ace left lowers it. The direction depends on what you saw, not on a general rule that seen cards make everything more likely.
If you know only that a card was burned and you did not see it, you do not get to use either revision. An unseen burn leaves your ace chance at 4/52 for any specific unseen card, by symmetry. Pretending you saw it is not a harmless shortcut.
What a condition is allowed to include
A condition is something you actually know. Your own two cards are a condition. A board that has been dealt is a condition. Another player's folded cards, if you did not see them, are not a condition. They are unknown cards still inside the count of unseen cards. Moving them into the given clause because you have a hunch will manufacture a sharp percentage out of a fog.
This is also why a streak of results on a wheel is a bad condition to feed into a deck formula. The wheel did not remove pockets. The right model for that device is the independent trial in why a coin flip has no memory. Use the conditional update on objects that stay out once they appear.
A one-line practice
Before you state a revised chance, finish this sentence: I am conditioning on, and I know it because I saw it. Then recount the remaining cards. If the sentence names a guess, stop. The original unconditional chance is the honest number until real information arrives.
The recount is a study skill for adults. It does not turn a card game into a favorable purchase, and local law still sets the age and the legality.
Frequently asked questions
After one ace is dealt, what is the chance the next card is an ace?
3/51, or 1/17, from a standard deck, provided no other cards have left and the first ace is really known.
Does an unseen burned card change that chance?
Not for practical purposes. If you did not see it, you cannot condition on its rank, and symmetry leaves the ace chance where it started for the cards still unseen.
Is 1/17 the chance of being dealt two aces?
No. 1/17 is the chance the second card matches after the first was an ace. The chance both are aces, from the start, is 1/221.