Cards Drawn in Increasing Order
A deck contains cards labelled through . You draw four cards one at a time without replacement. Find the probability that the four cards you draw appear in strictly increasing order of their labels.
Work it out, then compare with the answer.
SolutionShow the worked solution
Key idea. Once you know which four labels were drawn, every order in which they could have appeared is equally likely.
Each ordered sequence of four distinct labels is equally likely to be the sequence drawn. There are such sequences.
Group these sequences by their underlying set of labels. Each -element set gives exactly ordered sequences, all equally likely. Exactly one of them, , is strictly increasing. Ties cannot occur because the labels are distinct.
So, conditional on the set drawn, the probability of increasing order is . This does not depend on the set, so it is also the unconditional probability.
Equivalently, the increasing sequences correspond one-to-one with -element subsets, which gives
The deck size plays no role.
Variations
The same idea with a twist. Each solution gives only the changed step.
A deck has cards, each labelled with a distinct value from to . A dealer shuffles the deck and deals the top five cards face up in order. You pay $1 to play, and you receive if these five cards are either strictly increasing or strictly decreasing, and nothing otherwise. What must be for this game to be fair?
Work it out, then compare with the answer.
Solution
The five dealt cards have distinct labels in uniformly random relative order, so all orders are equally likely. The deck size is irrelevant.
Exactly two orders win: the increasing one and the decreasing one. The win probability is therefore .
The game is fair when the expected payout equals the $1 stake: