Drawing Two Cards of the Same Colour From a Shuffled Deck

Two cards drawn without replacement from a shuffled deck match in colour with probability 25 over 51, just under one half.

Two cards are drawn without replacement from a well shuffled standard 52 card deck, 26 red and 26 black. The probability that both cards share the same colour is 25 over 51, about 0.49, just under one half. That value falls out immediately once the first card is fixed and only the second card's colour is left to condition on.

A standard deck has 26 red cards, hearts and diamonds, and 26 black cards, clubs and spades. Two cards are dealt one after another without replacement, and every ordering of the 52 cards is equally likely. The question is the probability that the two drawn cards are the same colour, either both red or both black.

Every possible arrangement of the 52 cards is equally likely once the deck is well shuffled, which means every pair of positions, first and second card, is equally likely to hold any two distinct cards from the deck. That uniformity is what allows the conditioning argument below to fix the first card without loss of generality.

Condition on the colour of the first card rather than splitting into two separate cases. Whatever colour the first card turns out to be, 25 cards of that same colour remain among the 51 cards left in the deck, since one card of that colour has already been removed. The second card matches the first card's colour with probability 25 over 51, and that value does not depend on which colour the first card happened to be.

The same answer follows from adding the two cases directly. The probability of two reds is 26 over 52 times 25 over 51, and the probability of two blacks is the identical 26 over 52 times 25 over 51 by symmetry, and the sum of those two equal terms is 25 over 51, matching the conditioning argument.