🖍 Color Switches

Published on 2nd September 2021
3 minutes read

I was going through a puzzle on brainstellar.com and something caught my eye. For sake of honor, this is a link to the page, here's how it went:

You are given an urn with 10 balls (5 black and 5 white). You pick balls from urn one by one without replacements until all the balls are out. A black followed by a white or a white followed by a black is "a colour change". Calculate the expected number of colour changes if the balls are being picked randomly from the urn.

Initially the problem appears daunting because there could be 10C5 (3,628,800) possible arrangements with these 10 balls. Counting the number of color changes in all of them seems impossible. And frankly, original puzzle had 50 balls each, so that definitely doesn't seem to be within counting range at all. But there exists a trick!

I have seen such problems before... and truth be told, I have never been too comfortable around the obtained theoretical results. What got me thinking was the question - How would the results look like if someone actually performed the experiment ?