Sometimes it seems impossible to improve on a random choice. If you know nothing about two numbers, choosing one of them should give you exactly a 50% chance. This puzzle reveals a very fine crack in that intuition.
The Larger of Two Envelopes
Riddle statement
Someone writes two distinct numbers in two envelopes.
They may be small, huge, positive, negative, close together, or very far apart. You know nothing about them.
You choose one envelope at random and look at the number inside.
Then you may either keep that number or switch to the other envelope.
Is there a strategy that gives you a probability strictly greater than 50% of ending up with the larger number, no matter which two numbers were written?
Show solution
Solution
Yes.
Before opening an envelope, randomly generate a reference number $R$ using a distribution with support over the entire real line, for example a normal distribution. This guarantees that every real interval has a positive probability of containing $R$.
Open one envelope and observe the number $x$:
- if $x>R$, keep the envelope;
- if $x<R$, switch to the other one.
Let $a<b$ be the two written numbers.
- If $R<a$, the strategy is equivalent to always keeping the first envelope, and it succeeds with probability $1/2$.
- If $R>b$, it is equivalent to always switching, and it also succeeds with probability $1/2$.
- If $a<R<b$, it succeeds with certainty: you switch when you see $a$ and keep the envelope when you see $b$.
Since the interval $(a,b)$ is not empty and the chosen distribution assigns it positive probability, there is a positive probability of entering the certain-success case. In all other cases, the strategy remains at $1/2$.
Answer: yes. An independent random reference with support over the whole real line lets you beat 50% strictly for any pair of distinct numbers.