interesting but simple problem

[borisp]

Maybe this problem has been posted before, but here it is anyway. It is a basic example of more general principles involved in the theory of quantum computing; I found in on John Baez's blog (a physics prof at one of the UC's).

You and a friend are each going to flip a fair coin, simultaneously. You each get to look at the result of your flip, and then you each have to guess about the outcome of your friend's flip. No communication is allowed during the whole process, but you can discuss strategy beforehand. You "win" if you both guess correctly, otherwise you "lose." What is the best probability of "win" that you can achieve?

(Specifically, I am posting this problem because it tangibly illustrates the difference between the notions of "guessing the outcome" and "probability." The confusion between the two seems to be the origin of the Sleeping Beauty "paradox.")

[m_the0ry]

(1/2)^(1/2) by Bell's Theorem.

[borisp]

wrong, but in a very sophisticated way (honestly)

[PairTheBoard]

You each guess the other's coin will land the same as yours - which you know. That way, you will each guess right when your coins match, which happens half the time. As a team you are employing a strategy which produces a correlation between your two guesses. In other words, your guesses are no longer independent. You will be guessing the same when your coins are the same, which is favorable to the team.

PairTheBoard

[borisp]

Yes!! or you guess the opposite of your own result, either way you get 50% win rate.

Further, this is obviously an upper bound to your win rate, since you must get at least one guess right, and that happens no better than 50% of the time (by the theory of probability...)

[oldbookguy]

[ QUOTE ]
You each guess the other's coin will land the same as yours - which you know. That way, you will each guess right when your coins match, which happens half the time. As a team you are employing a strategy which produces a correlation between your two guesses. In other words, your guesses are no longer independent. You will be guessing the same when your coins are the same, which is favorable to the team.

PairTheBoard

[/ QUOTE ]

I would put your odds at 25% if using strategy you both decided in advance how you will answer each time or if you guess the other coin matches yours.

Consider:

Coin A Coin B
Heads Heads
Heads Tails
Tails Heads
Tails Tails

Yes, the odds of a coin being heads / tails is 50% but each has its own 50% that must be considered as well.

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
You each guess the other's coin will land the same as yours - which you know. That way, you will each guess right when your coins match, which happens half the time. As a team you are employing a strategy which produces a correlation between your two guesses. In other words, your guesses are no longer independent. You will be guessing the same when your coins are the same, which is favorable to the team.

PairTheBoard

[/ QUOTE ]

I would put your odds at 25% if using strategy you both decided in advance how you will answer each time or if you guess the other coin matches yours.

Consider:

Coin A Coin B
Heads Heads
Heads Tails
Tails Heads
Tails Tails

Yes, the odds of a coin being heads / tails is 50% but each has its own 50% that must be considered as well.

[/ QUOTE ]

When H-H you both look at your own coin, you both guess the other has heads, and you are Both Right.
When T-T you both guess the other has Tails and you are Both Right.

So you Both guess right 50% of the time. If you were both guessing independently and randomly you would each be right 50% of the time and by Independence, you would Both be right (.5)(.5) = 25% of the time. The strategy has improved your chances of "Both Being Right" from 25% to 50%.

PairTheBoard

[chillrob]

While this strategy would work, I don't think it matches the original terms of the problem, which stated "you each have to guess about the outcome" of the other flip.

If you follow this strategy, you are not following my definition of a "guess", which would be to give your gut intinct as to what the result of the other flip was, either based on some information you may have, or the lineup of the stars, or the feeling in your stomach, whatever. Simply naming the the result of your flip (or the opposite) is not really a "guess".

Rob

[kerowo]

Why can't this strategy be incorporated into your gut? If not then there is no way to get above random chance here.

Congratulations though, in some parts I'm considered a bit of a nit but this is nittiness raised to a new level.

NH Sir.

[borisp]

"Guess" here refers to the announcement of a choice, without true knowledge of whether your choice will be right or wrong. Sorry if this was confused with the everyday notion of "guess," which might invoke the idea of randomness...