CoachA symmetry argument computes a probability or an expectation by exhibiting a transformation of the sample space that preserves probabilities and exchanges the outcomes of interest, so that they must have equal probability, without computing either.
Examples
Example 10.6 (The first ace)
Where, on average, is the first ace in a shuffled deck? The four aces cut the 48 other cards into five gaps, and by symmetry each gap holds on average cards, so the first ace is at position on average. An exhaustive count over small decks confirms the formula .