Beach Coin Standoff
You and your brother are sent to separate rooms and cannot communicate. Each of you independently decides whether to walk out carrying a coin. The coin is biased: it lands heads with probability 0.6.
- If both of you carry a coin out, your parents flip both, and you go to the beach only if BOTH land heads.
- If exactly one coin comes out, they flip that one, and you go if it lands heads.
- If neither of you brings a coin, you stay home.
Before being separated you agreed on one rule: each of you carries the coin out with probability , independently of the other. Choose to maximise the chance of going to the beach.
What is that maximum probability?
Answer format: round to 3 decimal places.
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