Escape From Evil's Room

An evil wizard places 10 people in a room and forces them to play the following game.

He places on each person’s head either a red hat or a blue hat independently with probability \(\dfrac{1}{2}\). Each person can see the colors of the hats of all other 9 people, but not the color of his own hat. Simultaneously, each person must say a real number.

They win if

  • The sum of the numbers they say is strictly positive and there are an even number of red hats, or
  • The sum of the numbers they say is strictly negative and there are an odd number of red hats.

If these 10 people can decide on a strategy beforehand, find their maximum probability of success, and let this value be denoted as \(P\).

Submit your answer as \(\left\lfloor 10^4\times P\right\rfloor\).

Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.


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