So you're about to partake in a gruesome series of death games for the amusement of the rabble, or you're entering in a contest to see if you can beat out others to win a hugely expensive artificial organ, or you're about to be publicly executed for being in possession of a Guy Fawkes mask. There are plenty of ways to be done away with by the state in a macabre, carnivalesque celebration, given the many dystopian futures that fiction has warned us about.
Anyway, before you die, you play this torturous little game. Contestants are presented with a set of three cups. Under one is a pardon. Under two are nothing but dashed dreams. After the contestant has picked a cup, without seeing what is underneath it, one of the others (always an empty one) is raised. That leaves two cups; the one the contestant first picked, and the one that wasn't raised. The contestant then has to decide whether they want to switch cups or stay with the one they first picked.
At first glance, the odds seem the same either way.
Each of the three original cups had a one-third chance of being the winning one. With one eliminated, each remaining cup is as likely as the other to have the winning ticket underneath. You have a fifty-fifty chance either way.
Actually, you double your chances by moving to the other cup. The odds aren't split between the two cups, they are all piled on the new cup, which gives you a two-thirds chance of winning while the cup you originally picked gives you a pitiful one-third. This may seem counter-intuitive. The best way to understand it is not to calculate your odds of picking the right cup, but your odds of picking the wrong one.
If there are three choices, and you can pick one, you have a two-thirds chance of getting it wrong. This means that when the sadistic game show host, possibly played by William Shatner, raises up one of the other cups, eliminating the loser, there is a two-thirds chance he is 'showing' you the winning cup. Meanwhile the original cup you picked still has one-third odds of being the winner. You're sitting on losing odds and staring at winning ones.
And if it turns out that you beat the odds on the first try and got the cup with the winning ticket only to lose when you switched, you can go to your death knowing you played a solid technical game. Well done!
Via Cut the Knot.