medium | probability |

A and B are in a team called AB, playing against C. If AB team wins they win Rs 3, nothing otherwise.

The Game: A and B are placed in 2 separate rooms far away. A will toss a coin and B will also toss a coin; A will have to guess the outcome of B's toss and B will guess A's. If both guesses are right, team AB wins Rs 3, nothing otherwise. Should they play the game, by paying Rs 1 at the start?

The winning probability is not 1/4. They can make a strategy before the game.

1/2

Let's list down all the possibilities.

A | B |
---|---|

H | H |

H | T |

T | H |

T | T |

Observe that $2$ out of $4$ times, their coins have the same face, i.e. either both heads or both tails.

They can speak their own coin's face as their guess. They win the game with a probability of $1/2$.

The pay-off will be positive ($1/2 \cdot 3 - 1 = 0.5$), and hence they should play this game.