Symmetry: when relabeling does the counting for you
If two events are interchangeable by relabeling, they have the same probability. Saves you from the case-bashing interviewers love to test.
Method · Symmetry
Intro
When a setup is invariant under some relabeling — swap two cards, permute the positions, rotate the circle — events that map to each other under that relabeling have equal probability. That’s the whole technique. The payoff is that you almost never need to enumerate: a one-line bijection or an “each of $n$ slots equally likely” observation collapses the problem. The cue is a uniform-random structure (shuffle, permutation, random seating) and an event phrased in terms of which labeled objects land where.
✓ Intro · expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine — being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
✓ Try first · expand
Worked example
A standard 52-card deck is shuffled uniformly at random. What is the probability that the ace of spades appears before the ace of hearts in the shuffled order?
✓ Worked example · expand
Practice 1 of 3Type a fraction, decimal, or expression — mathjs parses it.
✓ Practice · expand
Reflection
When you read a probability problem, what’s the cue that tells you a symmetry argument will work — and what’s the cue that it <em>won’t</em> (i.e., when the setup looks symmetric but the event you’re asked about secretly depends on something the relabeling doesn’t preserve)?