## Saturday, April 25, 2009

### Game show puzzle: we have a winner!

A few weeks back, I posted my favorite statistics puzzle... and Justin Horn is the winner of the, er... figure it out and post the right answer in the comments... contest. Sort of.

Not really a contest, but Justin got it right.

You can do the gag yourself with cards. Take three cards; two "2's" and a king (or a queen, your call... or an eight, if you're, well, *that* kind of person). Have a friend hold them up. You pick one, and he puts down one of the other "2's." You can then stay with your choice or switch. Do it each way a couple dozen times (I have). You will find that switching gets you the "good" card two times out of three.

This seems, of course, contrary to sense. That's the whole point of a puzzle, eh? You have a choice of two things. Stay with one or switch... sould be 50/50, right? Well, as Justin points out... no.

Your original choice was a 1/3 chance of getting the right card. If you elilminate one of the other bad choices, and switch, you've converted your chances to 2/3. Still not convinced?

Imagine this. Same game show, but the host has you pick from 100 doors. He then shows you crap behind all but two of them. Does it make sense to stay or switch? He's opened 98 doors of crap. What are the odds that the door you originally picked is the good one? That's right... 1%. What are the odds that the other one is the right door? 99%. Is it a 50/50 shot now?

I love this stuff.

BONUS PUZZLE: Imagine a string of lights stretching off into infinity, numbered 1 to infinity. Each has a trigger that, when touched, will switch its state. If the light is off and is touched, it turns on. If it's on, it turns off. So... an infinite number of bunnes lines up behind light #1, and the first one goes and jumps on every single light, turning them all on. Bunny number 2 jumps on every other light. Bunny number 3, on every third. Bunny 4, on every fourth. You get the pic... the bunnies jump on multiples of the light based on their place in line. QUESTION: Describe the set of lights that is on after every bunny takes his or her turn.