Cards Without Replacement Interview Questions
How to solve card probability questions where draws change the deck and independence is not available.
Candidates who accidentally assume independence in card and sampling prompts.
The deck changes
Without replacement means every draw changes the number of remaining cards and often the number of favorable cards. The second draw usually does not have the same probability as the first.
Use sequential probabilities
For ordered draws, multiply conditional probabilities. After an ace is drawn, there are three aces among 51 cards. After a non-ace, there are four aces among 51 cards.
Use combinations for hands
For unordered hands, combinations are often cleaner. Exact counts like "two aces in five cards" become C(4,2)C(48,3)/C(52,5).
Concrete example
The chance the second card is an ace can be found by symmetry as 4/52. But if the first card is known to be an ace, the chance the second is an ace becomes 3/51.
Conditional information
Information about previous cards changes the deck. Information about the hand as a whole changes the sample space. Say which type of information the prompt gives.
Common mistakes
Candidates often multiply 4/52 twice for two aces. That assumes replacement. Without replacement, the deck state must be updated after each draw.
Practice the pattern
Use the LeetQuidity curriculum and calibration to turn this topic into a focused practice plan.