Counterfeit money, what's the optimal strategy here

Bob and I found two 50 dollar bills out of nowhere. We know they're either both legitimate or both counterfeit. If they're legitimate, they're worth 50 dollars each, otherwise 0. I get one 50 dollar bill, Bob gets the other 50 dollar bill.

Bob offers to buy my 50 dollar bill for 10 dollars or sell his bill to me for 10 dollars. This is before it's revealed whether they're both legitimate or counterfeit.

An online gambling service lets me make a one-to-one bet that the 50 dollar bills are legitimate or counterfeit. For example, if I bet 60 dollars that the two bills are counterfeit and both turn out to be legitimate, then I lose 60 dollars. But if I bet 60 dollars that the two bills are legitimate and both turn out to be legitimate, I earn 60 dollars.

So my options are:

- Buy Bob's 50 dollar bill for 10 dollars.
- Sell my bill to Bob for 10 dollars.
- Bet that the bills are legitimate for C amount with one-to-one odds.
- Bet that the bills are counterfeit for C amount with one-to-one odds.

If the bills end up being legitimate (we won't know this until afterwards), they're worth 50 dollars each, otherwise 0.

QUESTION:
What is my optimal play here, to maximize the amount of money I'm guaranteed to win?

MY THOUGHTS:
Buy Bob's 50 dollar bill for 10 dollars, and then bet 25 dollars on both bills being counterfeit.

Why bet 25 dollars on both bills being counterfeit? The profit in the two cases:

- Both legitimate: −10 + 50 - C = 40 - C
- Both counterfeit: −10 + C

Theoretically, any value of C such that 40 - C > 0 and -20 + C > 0 will generate a guaranteed profit no matter the outcome, but I want to make the same amount of money no matter the outcome. We don't know what are the respective probabilities that they're both legitimate or both counterfeit.

So let's set the two equations equal: 40 - C = - 10 + C, which implies that C = 25. So my strategy guarantees a profit of 15 dollars no matter what if we bet 25 dollars on both bills being counterfeit (and also buy Bob's 50 dollar bill for 10 dollars).

WHAT I'M CONFUSED ABOUT:
In my two equations above, what if instead considering the profit in both cases, we instead use the total amount we end up with:

- Both legitimate: -10 + 100 - C = 90 - C
- Both counterfeit: -10 + C

Setting the two equations equal gets us C = 50, which guarantees we end up with 40 dollars total no matter what if we bet 50 dollars on both bills being counterfeit (and also buy Bob's 50 dollar bill for 10 dollars).

So which method is correct, the one considering total profit or the one considering the total amount we end up with?

You should maximize your ending amount of money.

Interesting question.

I would gladly pay $10 for it. Even if it is fake I am pretty sure I could find someone to buy it from me for more than $10. There are lots of stupid people out there.

Quote: DRich

I would gladly pay $10 for it. Even if it is fake I am pretty sure I could find someone to buy it from me for more than $10. There are lots of stupid people out there.
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I don't think I'd want to get mixed up with the kind of people who are in the market for counterfeit $50 bills.

I'd just sell mine and take the $10. Easy profit. No downside.

I picked up a Thomas Jefferson dollar coin at the bank recently and there was no date on it. I thought it was counterfeit but I took it so I could look it up. There's a market for some of these where they are worth 100's or 1000's of times the $1 value.

The bank was going to charge me 2% for any checks cashed over $50 even though the check was made on their bank. They gave me a break because the check was barely over $50, but next time I won't be so lucky! I saw somebody else there getting Canadian money but I figure only customers of the bank can get the currency changed out.

People trying to sell $1 coins for $3.
COMPLETE Presidential Dollar full Set Brilliant Uncirculated 40 Coins BU Mint!
https://tinyurl.com/2bjqjb8s

Quote: ChumpChange

People trying to sell $1 coins for $3.
COMPLETE Presidential Dollar full Set Brilliant Uncirculated 40 Coins BU Mint!
https://tinyurl.com/2bjqjb8s
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It's for coin collectors. If they're really BU Mint they're worth more than face value.

Quote: ChumpChange

I picked up a Thomas Jefferson dollar coin at the bank recently and there was no date on it. I thought it was counterfeit but I took it so I could look it up. There's a market for some of these where they are worth 100's or 1000's of times the $1 value.
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Are you talking about those Presidential dollar coins? None of them have a date on them, unless you're referring to the years Jefferson served as President??

Why would I think a coin with no date on it wasn't counterfeit? It was probably a 2007 or 2008. It might as well be a token for a nearby arcade.

Find a real 1975 quarter and you'll make out well.