Points on a ring
Thinking about my undergrad days studying math, I wish more problems were visualized like this
Points on a ring N= 4 ✗ N= 5 ✗ N= 3 ✓ N= 6 ✓ N= 4 ✓ N= 5 ✗
Here's a problem that shows up in math competitions and quant interviews:
Drop 4 points randomly on a circle. What are the chances they all land in the same half?
Drop 4 points click drop to place points
Try it a few times. Sometimes all four cluster into one semicircle, sometimes they spread out. What probability would you guess?
The obvious (wrong) answer
The circle is symmetric, so place one point anywhere and ask whether the other three land in the same semicircle. Each of those 3 points independently has a 1 / 2 1/2 1/2 chance of landing in that half, so the probability should be ( 1 / 2 ) 3 = 1 / 8 = 12.5 % (1/2)^3 = 1/8 = 12.5\% (1/2)3=1/8=12.5%.
Test that reasoning here. Click on the circle to fix your point, then drop 3 more and see whether they land in your semicircle. Repeat it many times and watch the rate:
... continue reading