Coin Tossing Game Optimal Stragegy A and B are playing coin toss game with a fair coin as a team, both of them have to be correct in a game to be able to gain one point, no communications will be allowed after the game starts.
Assuming A has some super power and knows all of the results after the game starts, to be more specific, A knows what the current toss will be and what the next result will be. However, B is just an ordinary person.
Assuming they are going to play this game forever, is that possible A and B could come up with a strategy before the game starts so that the possibility of their correct guess is greater than 0.5 in the long run?
 A: This answer assumes A knows all of the results after the game starts, or more specifically, A knows what the current toss will be and what the next result will be.
B will select heads (arbitrary) on the first toss and then follow the rule established before the start of the game.
Rule: If B incorrectly guesses the current toss, then B will guess on the next toss whatever A guessed on the current toss.
A and B will use a predetermined strategy to capitalize on A’s ability. An obvious pattern would be for B to guess the same outcome as the previous toss unless directed differently by A from the Rule.
Example 1:
Toss 1 will be heads.
A knows B will select heads and they both guess correctly.
Toss 2 will be tails, but A also can see toss three will be tails. A knows B will incorrectly guess heads, so A signals Toss 3 will be tails by guessing tails via the Rule.
Toss 3 will be tails.
A knows B will select tails because of Rule from Toss 2 and they both guess correctly.
Example 2:
Toss 1 will be heads.
A knows B will select heads and they both guess correctly.
Toss 2 will be tails, but A also can see toss three will be heads. A knows B will incorrectly guess heads, so A is signaling Toss 3 by guessing heads via the Rule.
Toss 3 will be heads. A knows B will select heads because of Rule from Toss 2 and they both guess correctly.
Notice how in these examples, the team cannot possible miss two tosses in a row.
If my thinking is correct...
The subset for two tosses changes from {CC, IC, CI, II} of equal distribution to {CC, CI, IC}, where C denotes correct and I denotes incorrect.
All of the II become IC so the distribution is CC=0.25, CI=0.25, IC=0.5 for very large numbers of games.
CC x 0.25 outputs 0.5C
CI x 0.25 outputs 0.25 C and 0.25 I
IC x 0.5 outputs 0.5 I and 0.5 C
Total ratio is 1.25:0.75 or 5:3 Correct to Incorrect. vs 1:1 by a no strategy method.
Conclusion: long run success = 0.625
