Symmetry

When outcomes are interchangeable, their probabilities are equal.

What is it?

If relabeling players, positions or values leaves the problem unchanged, the corresponding outcomes have equal probability.

Symmetry often gives the answer in one line where direct computation would take pages; interviewers reward spotting it.

Think of it when the setup looks the same after swapping two participants, items or positions.

Think of this method when…

Identify the interchangeable outcomes; if there are $m$ equally likely ones and exactly one is favorable, the answer is $1/ m$ .

P (X_{1} is the largest of X_{1}, \dots, X_{n}) = \frac{1}{n} (i.i.d. continuous)

1Find the transformation (swap, reflect, relabel) that leaves the problem unchanged.
2List the outcomes it maps into each other.
3Set their probabilities equal and use that they sum to 1.
4Handle any leftover asymmetry separately.

A flips $n + 1$ fair coins and B flips $n$ . What is the probability A gets more heads than B?

1
Compare B with A's first $n$ coins: by symmetry $P (A_{n} > B) = P (A_{n} < B) = p$ , and $P (A_{n} = B) = 1 - 2 p$ .
2
A wins if $A_{n} > B$ , or if $A_{n} = B$ and the extra coin is heads.
$p + \frac{1}{2} (1 - 2 p) = \frac{1}{2}$
Answer: $1/2$

A deck is shuffled. What is the probability the 17th card is an ace?

17 QuantSolve questions are tagged with this method.