June 03, 2017

A puzzle about inference

Here is a puzzle which I often use as a starter for a course of lectures on probabilistic coupling. The puzzle arose during some research -- I devised it as a simple example to show myself why a particular idea would not work -- and I developed the solution using calculations. In a lunch queue, I asked James Norris of Cambridge whether he could think of a calculation-free answer, and he found one almost immediately.

  • You go to Starbucks with a friend.
  • You are the quiet type, and stay in the front room near the window, sipping a coffee slowly to make it last.
  • She is extrovert, and has gone through to a further room where the louder people gather.
    There she plays a game: toss n coins, win if r heads or more.
  • After the game should have finished, a third friend comes through and says,
    either: "She won the first toss'';
    or: "She won at least one toss''.
    (Why don't people speak more clearly?!)
  • Which is better news for your friend? Prove your answer without calculation!

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