Three players enter a room. As each player enters, a coin is flipped (independently of the other players) and either a red or blue hat is placed on that player's head. The players can not see the colors of their own hats, only the colors of the other two players hats. Once everyone is in the room, each of the players will simultaneously guess his hat color or pass. If at least one player guesses correctly and none guess incorrect, the players will win a cash prize. There is no communication between the players once they enter the room, but they are allowed to discuss a strategy prior to entering. The naive strategy would be for one player to just guess red and the other two pass, giving a probability of winning 1/2. Is there a strategy that gives a higher probability? Hint: there is a strategy that yields 75%
There are 8 possible arrangements of the hats, each having the same probability of occuring (assuming a fair coin).
BBB BBR BRB RBB RRB RBR BRR RRR
The agreed strategy would be: If the other two players are wearing the same color, then guess the opposite color. Otherwise pass. This strategy fails for RRR and BBB only, but wins in the other 6 cases.
Yes, there is a strategy that yields a higher probability of winning the cash prize. In fact, it is possible to achieve a 75% probability of winning.
To understand this strategy, we need to consider all the possible scenarios and try to find a pattern. Let's denote the hat colors as follows: Red (R) and Blue (B).
First, let's analyze the possible scenarios:
1. All three players have the same color hat (RRR or BBB): In this case, any player can make a guess and it will be correct. Therefore, the players have a 100% chance of winning.
2. Two players have the same color hat and one has a different color hat (RRB or BBR): In this case, the player with the different color hat can deduce their own hat color based on the hats of the other two players. For example, if one player sees two red hats, they know their own hat must be blue. Therefore, the two players with the same color hat should pass, and the player with the different color hat should make a guess. This strategy guarantees a 100% chance of winning in these scenarios.
3. Two players have different color hats (RB or BR): This is the most challenging scenario. In this case, no player can immediately deduce their own hat color based on the hats of the other two players. However, they can make some guesses and pass the turns to give clues to each other.
To devise a winning strategy for this scenario, the players can agree on an order in which they will make their guesses. Let's designate the players as Player 1, Player 2, and Player 3.
Player 1 goes first and looks at the hats of Player 2 and Player 3. They count the number of red hats they see and makes their guess based on that count. For example, if Player 1 sees two red hats, they guess that their own hat is red. Otherwise, they guess that their own hat is blue.
Player 2 goes next and follows a similar approach. They look at the hat of Player 1 and either see a red hat or a blue hat based on Player 1's guess. Player 2 then counts the number of red hats they see (excluding the hat of Player 1) and guesses their own hat color accordingly.
Player 3 goes last and also follows the same procedure. They look at the hats of Player 1 and Player 2, determine the count of red hats they see, and make their guess.
This strategy ensures that at least one player will guess their hat color correctly, and no one will guess incorrectly. Thus, the players have a 75% chance of winning the cash prize.
To summarize, the strategy is as follows:
- If all three players have the same color hats (RRR or BBB), any player can guess, ensuring a 100% chance of winning.
- If two players have the same color hats (RRB or BBR), the player with the different color hat guesses, ensuring a 100% chance of winning.
- If two players have different color hats (RB or BR), the players follow the described order of guesses based on the counts of red hats they see from the other players. This strategy yields a 75% chance of winning.