Sunday, December 19, 2010
God's Wheel of Fortune
I'm sure all of my immediate family members were dragged in circles with this quandary at one time or another, and contributed their ideas, but I have to credit John Mark as the initial inventor. I recall trying to wrap my brain around this when I was about 6, so he was probably no more than 8 when he first posed this experiment.
It goes something like this:
You are given a wheel and a challenge. The wheel is just like the one from Wheel of Fortune, except it only has two segments, like a pie cut straight down the middle. It has a red half and a blue half. Also, this wheel belongs to God, so He can change it at whim. The challenge is simple: you must successfully spin the wheel and have it stop on the red segment, and you have an unlimited number of spins to get it right.
Here's the catch: every time you spin the wheel, God doubles the number of segments, and leaves only one red segment. So, after the first spin, the wheel is split in quarters, with three blue segmments and one red. After the second spin, the wheel is split into eighths (seven blue segments, one red). So forth and so on, ad infinitum.
Obviously, your odds of successfully landing the red segment are cut in half with each additional spin, but given that you have the rest of eternity to sit there spinning, are you guaranteed to eventually hit the ever-diminishing red sliver? (In some variations of this experiment, we would suppose that your admission to Heaven would be contingent upon successfully landing the red. In hindsight, this seems like a pretty cruel conception of God for two children our age to have possessed.)
Argument in favor: Yes. You are desinted to eventually hit the red segment. It doesn't matter how bad the odds get, whether it's one in a million or one in a billion. Even though the odds are doubling against you, every spin brings you closer to that inevitable fate - you must hit the red segment someday in the course of eternity.
Argument against: No. There is no guarantee that you will ever hit the red segment. As the odds get worse and worse, your hope of ever hitting the segment disappears. Throughout eternity, you have nearly no chance of hitting the red segment unless you get it within the first few spins, while the odds are still decent. It's going to be a long eternity in the awful Wheel of Fortune purgatory.
I've argued both sides before and never felt completely satisifed with either answer. So, with some time to kill on the airplane yesterday (as we were traveling home to visit Ryan's family), I posed this question to Ryan, who is a Economics major and has math practically seeping out of his pores. I was surprised and a little disappointed that he managed to wrap this question up with a bow within about a half-hour of first hearing it.
Apparently, this question which has boggled me for most of my life was not actually a universe-shattering paradox. I just lacked the know-how to calculate the probability. Dang.
I feel bad sharing Ryan's solution so immediately and depriving any readers the opportunity to tease this out themselves. But, I guess I'm a living testament that you could spend 15 years with this in the back of your mind and never get any closer to a solution.
I'm also so excited to finally have the real, mathematically-proven answer to this question that I just have to share. I encourage you to come up with your own hypothesis, fight about it with anyone who is willing to engage you, and then, come back if you really want to hear the answer, as math dictates it. For the answer, read on.
So, to John Mark and everyone else. The answer is: No. You are not guaranteed to hit the red segment.
Take us home, Ryan:
Assuming that spinning the wheel can be considered a completely randomized selection method, the odds of landing on the red segment on the first spin is one half. That's easy. The tricky part of all of this is figuring out what the odds are from the outset for landing a red on each successive spin. If we give it some thought, the odds of landing on the red on the second spin means that you did not land on the red on the first spin. So taking another spin should be considered as having odds of its own. So the odds of landing on the red on a successive spin must be the odds of getting the red on that spin multiplied by the odds of spinning that spin. So, for the second spin, the total odds of landing on the red segment are:
(1/2) + (1/2)(1/4)
The odds of getting it on spin 1 PLUS The odds of not getting it on spin 1 TIMES The odds of getting it on spin 2
The odds of landing on the red segment for the third spin are:
(1/2) + (1/2)(1/4) + ((1/2)(3/4)) (1/8)
The odds of not getting it on spin 1 nor on spin 2 TIMES the odds of getting it on spin 3.
For those of you who have taken a bit of calculus and are curious, this is called an infinite Taylor series and it is recrusive. This particular series approaches a finite sum (as you may have been able to tell, each additional spin adds a smaller and smaller increase to the total odds.) This will approach but never reach an asymptote of .70668 or 70.668%.
So, before your first spin, you have a 70.668% chance of ever landing the red segment. But, if you don't get it on your first spin (when your odds are 50/50), you are suddenly reduced to a 20% chance of ever succeeding.
Phewsh! Glad we took care of that one. Everyone can get down of the edge of their seats, and get back to their regularly scheduledly lived. We've put this one in the books.