# 1. A Counterintuitive Probability Problem

Mathematician Gil Kalai recently posted the following intuition-bending probability problem at his blog:

*You throw a dice until you get 6. What is the expected number of throws (including the throw giving 6) conditioned on the event that all throws gave even numbers.**

(*The problem is originally due to Elchanan Mossel. The term “a dice” is used here to denote a single die. Some people dislike this usage, though it does not obscure the question.)

Most people give the initially intuitive answer: three. But that’s wrong. I’ll get to the right answer below. Solutions, along with commenters’ expressions of bewilderment and contention, have been posted at Kalai’s blog, Math with Bad Drawings, and at Mind Your Decisions’ blog and YouTube channel. It’s interesting to note that one YouTube commenter accepted the correct solution, even giving an explanation in their own words, then later reverted back to the incorrect answer of three.

Methods for getting the correct solution are often hard to follow due to involving complicated-looking math or relying on background knowledge. I’ll share a solution that I think makes the problem more intuitive, and that only requires a basic understanding of probability, along with a little precalculus. Still, I’ll be sure to review the most relevant concepts as I go along, just in case it’s helpful. Continue Reading