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I need a mathematical derivation of Monty Hall problem. Where we repeat the game a times. Please derive the expected payoff for the policy with

I need a mathematical derivation of Monty Hall problem. Where we repeat the game a times. Please derive the expected payoff for the policy with P(switch) = a analytically. For every game, it is independent.

What value of a maximizes payoff?(We define the payoff of a single play of the game as 1 for the car and 0 for a goat)

Please send me answer ASAP.

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