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真假钞票

Mystery Bills

专题
Brainteaser / 脑筋急转弯
难度
L5

题目详情

你和朋友各有一张面额为 $100 的钞票。每张钞票可能是真钞，也可能是假钞，真假概率未知。你的朋友愿意以 $20 买下你的钞票，或者以同样价格把他的钞票卖给你。你还可以按 1:1 赔率，对两张钞票是真还是假投注任意金额，方向也可以任选。给定这些条件，你最多能保证赚到多少钱？

You and your friend each have a $100 bill. They could be real or fake with some unknown probability. Your friend will buy your bill for $20 or sell his to you for the same price. You can also bet any amount if the bills are real or not in either direction at a 1 to 1 payout. Given this, what is the most amount of money you can guarantee?

解析

如果只把自己的钞票卖给朋友且不做其他操作，可以保证赚到 $20，但还能做得更好。

考虑反过来以 $20 买下朋友的钞票。此时你手里有两张钞票。为了锁定无风险收益，需要在相反方向下注来覆盖两种可能情形。设我们下注 bb 美元，赌两张钞票都是假的。

若两张钞票都是真的，你拥有价值 $200 的钞票，但花了 $20 买入其中一张，并且下注亏掉 bb。此时收益为

180−b.180-b.

若两张钞票都是假的，钞票本身没有价值，你仍然花了 $20 买入朋友的钞票，但下注赢得 bb。此时收益为

b−20.b-20.

要保证无风险收益，应让两种情形下的收益相等：

180−b=b−20.180-b=b-20.

解得

b=100.b=100.

代回任一情形：

180−100=80,180-100=80, 100−20=80.100-20=80.

因此，通过买入朋友的钞票，并下注 $100 赌两张钞票都是假的，可以保证赚到

$80.\boxed{\$80}.

Original Explanation

If we sell our bill and do nothing we've guaranteed a profit of $20, but we can do better. Suppose you buy the bill for $20. The only way to guarantee a profit in this situation is to take a bet in the opposite direction to ensure your bases are covered. Let's say we bet bb dollars both that both bills are fake.

In the case both bills are real, you have $200 worth of bills but spent $20 acquiring one of them and a lost bet. Your earnings in this situation are 180−b180 -b.

In the case both bills are fake, your bills are worthless and you still spent $20 acquiring the one of them, but you made $b from your bet. Your earnings here are b−20b - 20.

The risk free situation is when you are making the same amount of money regardless of the outcome. Setting these equations equal to each we can see the bet amount to guarantee a return to be: 180−b=b−20180 - b = b -20 b=100b = 100 Substituting $100 back into either of our earnings formulas we get: 180−100=80180 - 100 = 80 100−20=80100 - 20 = 80 We see that by purchasing the additional bill and betting $100 both bills are fake we can guarantee a profit of $80