Coin Flipping Puzzle #1

50%. Agree to guess what you flipped.

If each prisoner uses the strategy of guessing the opposite of what they flipped, the chances that both will be correct is 50%... and that's the best they can do.

The only strategy I can see is based on the order of events. Since it's a 50/50 chance, and you will see your own result before making your prediction, I think your best chance is to predict the opposite result for the other prisoner (slightly better than guessing the same). Without that order of events, your combined chance is a straightforward 25%, but I think that condition increases your combined chance to 33 1/3%. Not that I can make a mathematical proof of it, but it feels logical.

There are 4 options for the 2 coins, HH,HT,TH,TT Once a prisoner has flipped he knows he can eliminate the pair of whatever he didn't flip as an outcome. 2 of the 3 remaining options are one of each so guess the opposite of your flip.

There are four possible outcomes with the two coins:
#1 = hh / #1 says: "I guess tail." #2 says: "I guess tail." (result: wrong wrong)
#2 = ht / #1 says: "I guess tail." #2 says: "I guess head." (result: right! right!)
#3 = th / #1 says: "I guess head." #2 says: "I guess tail." (result: right! right!)
#4 = tt / #1 says: "I guess head." #2 says: "I guess head." (result: wrong wrong)

And yes, CrystalMath's answer of guessing what you flipped (instead of guessing the opposite) works just as well.

You don't indicate what both flips are. Just what you think the other guy flipped. Just just "heads" or "tails."

I flunked algebra in 1963
because I read James Bond books
inside my algebra book instead of
paying attention.

Either both guess opposite or both guess the same. 50% chance of success.

I originally voted 25% based upon two random guesses.

Now, with strategy, I wanna say 50%.

If, with or without strategy, one chooses the same as what he flipped and the other chooses the opposite, then they will never both be correct.

Therefore, the strategy discussion will only be to decide if they are both going to say the same or opposite of what they each flipped.

Note that the instructions do not specifically state that you must show your flipped coin to the guards. Strategy:
Agree with your friend that you will each flip your coin, keep it hidden from the guards, and declare that you flipped heads. Then you each "predict" that your friend "flipped" heads. If you and your friend can get this strategy past the guards, then you have a 100% chance of winning.

Otherwise, I think your chances of survival are only 25% -- the probability that both of you can correctly guess the result of 50-50 coin flips.

Because I have two conflicting answers here, I have not voted.

The guards will ensure a fair flip, sure... but have you ever heard the story of the coin flipper that took Ken Uston for a lot of money? Would we be allowed when to decide to take this challenge? I would practice all day every day for a month straight flipping the coin, landing it in my palm and feeling which side was up and down. Once I can accurately tell which side it is, agree on an outcome (let's say Tails) and then ensure you flip tails. If you feel the tails is down on your palm, flop the coin over on to the back of your other hand, which is a completely legitimate coin flip technique. This would provide a 100% success rate.

Of course getting to the math question at hand though...

If 1 person were to do this and call the coin before they flipped it:
P(guessing 1 flip correctly) = 1/2 = .5 or 50%
P(guessing 2 flips correctly) = P(guessing 1 flip correctly) * P(guessing 1 flip correctly) = .5 * .5 = .25 or 25%

There's a lot more options here than I think people realize. The outcome of the coin flip is binary 00, 01, 10, 11... but that's not factoring your choices in to each outcome.

0000 (RIGHT)
0001
0010
0011
0100
0101
0110 (RIGHT)
0111
1000
1001 (RIGHT)
1010
1011
1100
1101
1110
1111 (RIGHT)

4/16 = 2/8 = 1/4 = .25 or 25%

Flip-Guess, Flip-Guess, Correctness
T - Heads, T - Heads, WRONG
T - Heads, T - Tails, WRONG
T - Tails, T - Heads, WRONG
T - Tails, T - Tails, RIGHT

T - Heads, H - Heads, WRONG
T - Heads, H - Tails, RIGHT
T - Tails, H - Heads, WRONG
T - Tails, H - Tails, WRONG

H - Heads, T - Heads, WRONG
H - Heads, T - Tails, WRONG
H - Tails, T - Heads, RIGHT
H - Tails, T - Tails, WRONG

H - Heads, H - Heads, RIGHT
H - Heads, H - Tails, WRONG
H - Tails, H - Heads, WRONG
H - Tails, H - Tails, WRONG

Now after I flip, let's say Tails, then I can eliminate the 2 groupings of 4 where my flip was Heads. I'm down to 8 possibilities. If I tell my partner I'm always going to guess the opposite of what I flip, then that means I'll be guessing Heads in this scenario. This puts me down to 4 possibilities, 3 of which are wrong and 1 which is right. Because even if my partner does flip Heads, he may guess my coin incorrectly. The 4 resulting options where I flip tails and guess Heads are:
T - Heads, T - Heads, WRONG
T - Heads, T - Tails, WRONG
T - Heads, H - Heads, WRONG
T - Heads, H - Tails, RIGHT

ERROR: I did not take in to account if my partner ALSO agrees to always guess opposite, then that eliminates 2 more of the choices leaving only 2 choices (one being right) for a 50/50:
T - Heads, T - Heads, WRONG
T - Heads, H - Tails, RIGHT

Therefore the answer would be 50% if you both either A) agree to always guess the same as you flipped, or B) agree to always guess the opposite of what you flipped.

The way I see it is that the best they could do is 50%.

Let's assume that they agree to simply predict that the other coin will be a match. For example, Prisoner A flips a head and predicts then that Prisoner B will also flip a head. Assuming a fair coin and a fair flip, he has exactly a 50% chance of being correct. If Prisoner B flips a head he will also predict a head and they will match. If Prisoner B flips a tail then he will predict a tail and they won't match. Regardless of Prisoner A's flip, he will predict Prisoner B's flip correclty 50% of the time.

I can't see how they can do better than that.

As far as I know, there are two strategies which will work 50% of the time:

1. Both predict the same as what they flip (will work with TT and HH).
2. Both predict the opposite as what they flip (will work with TH and HT).

Person 1 always predicts the same of his toss.
Person 2 always predicts the opposite of his toss.