When probability runs into logic?

[ElSapo]

"I hold AK, so the chances of him having AA are diminished."

I see this thought, or something akin to it, and I have this thought, or something akin to it, quite often. But I wonder if it's correct.

What I mean is this - does my holding an Ace actually lower the chances of someone else holding Aces? or any Ace?

It seems like a no-brainer "of course," but then I started thinking about it like this - deal 10 hands out. The possibility of one of them holding AA is XXX (I don't know what it is). I then look at one hand and find AK. Does that lower the probability of AA being out there?

[BruceZ]

The possibility of one of them holding AA is XXX (I don't know what it is). I then look at one hand and find AK. Does that lower the probability of AA being out there?

Yes, it lowers the probability by almost half. Before you looked at your cards, the probability of each player holding AA was 6/1326, and the probability that one of you opponents had AA was very close to 9*6/1326. After you look, there are now only 3 ways to make AA out of 1225 possible hands, so the probability that AA is out very close to 9*3/1225.

[ElSapo]

I guess what's weirding me out is the idea that the probability of something that's already either happened or not happened is being changed by knowledge of something else that happened.

I mean, the cat in the box is either dead or alive, right?

[BruceZ]

Probability is always relative to the knowledge that you have. If you were sweating all the players, the probability that AA is out would be 0 or 1. If you're online drawing at a flush, the probability of making it is 1 in 3, but the computer already knows if you will make it or not, because it knows what the next 2 cards are. You don't have that knowledge, so it's 1 in 3 for you. What it means is that out of all the times that you are in that same situation, that is the percentage of the time it will happen. Your opponents will hold AA a certain percentage of the time. If you only consider the times you hold AK, they will hold AA only about half as often as when you consider all the hands.

As for Schrödinger’s cat, that depends on your interpretation of quantum mechanics. By the strictest interpretation, the cat is neither dead nor alive until you open the box and check. More precisely, it is actually in a superposition of states of being both dead AND alive at the same time. No one argues that this is a valid interpretation for subatomic particles, but some argue that a macroscopic being like a cat should not have this property. Some of them believe that gravity will cause the wave functions to collapse for macroscopic objects. Amazing experiments are underway to resolve this. Believe it or not, the first interpretation that the cat is really in a superposition of states, as bizarre as that may seem, is beginning to show experimental evidence of being the correct interpretation.

In essence, quantum theory says that the universe is fundamentally probabilistic because there are certain things that cannot be completely known. Those who oppose the strictest view, and Einstein was one, are essentially arguing that this probabilistic nature actually results from our incomplete knowledge of physics, and in the future a more complete view of the universe may provide the additional knowledge needed to make the universe deterministic again. For example, some favor the "many-worlds" view of quantum mechanics, where all possibilities occur in different parallel universes, and we are only allowed to view one universe at a time. It should be stressed that the existing quantum mechanical view of the universe, as bizarre as its conclusions may seem, is extremely well confirmed by experimental evidence of many different kinds. Its results are used by engineers. It is as close to "proven" as a scientific theory can be.

[Cyrus]

"I guess what's weirding me out is the idea that the probability of something that's already either happened or not happened is being changed by knowledge of something else that happened."

Have fun :

Play the Monty Hall game (http://math.ucsd.edu/~crypto/Monty/monty.html)

[RiverMel]

I wouldn't tell you about Newcomb's Paradox (http://members.aol.com/kiekeben/newcomb.html). I'm sorry for causing you to waste the rest of your life trying to solve this.

[Zeno]

I won the first time. Car behind door #2. Did not switch. Went with first impulse - which is always right. Right? /forums/images/icons/grin.gif

[Cyrus]

"I won the first time. Car behind door #2. Did not switch. Went with first impulse - which is always right. Right?"

Nope.

/forums/images/icons/laugh.gif

[DPCondit]

Re: Newcomb's Paradox

Take the closed box only, if the alien could predict what I would do with certainty, he would know that I would prefer one million dollars and would only choose the one box.

Don

[SittingBull]