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In a game like Deal or no Deal, what is the formula the banker uses to give an offer to the contestant? The banker offers the contestant an amount of money to quit the game, the offer based roughly on the amounts remaining in play and the contestant's demeanor, the bank tries to 'buy' the contestant's case for a lower price than what's inside the case.

I am simply curious to understand how this work. Thanks!

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A fair offer would be the expected value of what's in the briefcase. That is you sum up the probability that they have some amount of money times that amount. If for example there are three cases left, $10, 5000, and 25000$, then the expected value of his case is

$$ \left( 10 \cdot \frac{1}{3} \right) + \left( 5000 \cdot \frac{1}{3} \right) + \left( 25000 \cdot \frac{1}{3} \right) = 10003.33$$

This is basically the average amount of the remaining briefcases; it's a standard gambling calculation.

This would be for a fair offer however. On the last briefcase, the banker usually offers something pretty close to fair. The rest of the time though the banker offers intentionally low amounts. This is for the same reason that they interview family members: It's in their best interest to have only one contestant per episode. It's fewer prizes and keeps an audience. So I don't know if there's a formula that they use to undervalue their offers. (Only some one who worked on the show would know.) But I suspect that it's a less precise science. That is a human producer picks numbers that he knows the contestant won't take, so it doesn't really matter how much it is.

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