The Monty Hall Paradox

Puzzle

You are participating in a game where the host has placed a prize in one of three covered boxes. You are asked to pick one of the three boxes. After picking it, but before opening the box, the host opens one of the other two boxes, which she knows carries no prize in it. She then gives you the option of switching your pick. Are you better off switching or not?

Solution

It is always better to switch, since your initial chance of picking correctly is . So the chance of picking correctly after switching is . 
Many people find this unintuitive, since it seems like one should have a chance. Part of this intuition stems from psychology: we are either reluctant to trust the game host or resistant to change. And part of this natural reaction is just plain wrong, constituting an abuse of probability theory. To make the situation more intuitive, let us imagine the game involved 100 boxes, instead of three. The host opens 98 other boxes, which she knows to be empty, after you first pick a box. Would you switch to the other leftover box? In this case, it should be obvious, even without a calculation, that it is in your best interest to switch. After all, it is very unlikely that your first pick was the right one!

Here’s an absurd example of how psychology (and the specious application of probability) can mislead us: As the Large Hadron Collider (LHC) was being built at the CERN laboratory, someone claimed that CERN had a 50% chance of destroying the Earth by creating a black hole in the collider. That argument was based on the following spurious reasoning: Either the LHC will destroy the Earth or it won’t, so it is a 50/50 chance either way. ¹ 

Puzzle

Can you draw a closed curve on the plane such that no square can be inscribed in it? 
 

Solution