# An interesting bet

Someone comes to you and says: 'I bet you £1, that if you give me £2 I will give you £3 in return. Do you accept my offer?' What do you respond to him?

• any reasoning for the change? sounds like an opinionated question if that is the case which it seems like. – FishingCode Jul 31 '20 at 10:00
• I did it because it was redundant and removing it would make it simpler. – Curiousmind Jul 31 '20 at 10:02
• it actually changes the meaning of the question in a way, since the word promise is eradicated. – FishingCode Jul 31 '20 at 10:04
• I would take the bet and then never give them £2 – hexomino Jul 31 '20 at 13:56
• @hexomino "if a, then b" is true whenever "a" is false, so you would just be out a dollar. – AlexanderJ93 Jul 31 '20 at 20:32

The statement is quite confusing, so simplifying it is a must. So, after some rejigging -

If you give me £2, I may give you £3. The bet is £1 that this will happen

I've substituted "promise" for "may", because it's essentially the same thing, but makes more sense when dealing with probabilities. The case is whether you trust them or not:

You trust them

Well done you, now let's see the consequences:

1. You give them £2
You: -£2
Them: £2

2. They give you £3
You: £1
Them: -£1

3. They win, so you give them £1
You: £0
Them: £0

That was pointless!

You don't trust them

Erring on the side of caution are we? Let's see what could happen:

1. You give them £2
You: -£2
Them: £2

2. They break their promise, and don't give you £3
You: -£2
Them: £2

The statement did not happen, so they give you £1
You: -£1
Them: £1

Oh dear, you could lose £1 as well

You skew your own chances on purpose

The other alternative is that you don't give them the £2 to start with, although you'd seem pretty stupid to do this

1. You don't give them £2
You: £0
Them: £0

They don't give you £3, because it's dependant on you giving them £2
You: £0
Them: £0

You did not induce the conditional statement, so nothing else happens
You: £0
Them: £0

This seems strange, because you'd think you'd have to give them £1, but you never chanced it. A clear example makes this obvious

I bet you £10 that if Red Rum comes first, Lightning will come second

If the condition (Red Rum comes first) never occurs, the bet has no pay-out

As there is no way you can possibly make money on this, no matter how truthful they are, you should NOT take the bet

• This is a very complete answer, but wouldn't the bet winners go in the reverse order? I mean, in the case where you opt to trust them you would actually lose the bet. Isn't that so? – Curiousmind Jul 31 '20 at 14:48
• Whoops - I'm thinking of poker betting (both parties bet the same amount). I'll adjust my answer – Oliver Jul 31 '20 at 15:40
• Of course there are at least two other possibilities, they might give you more than £3 back . Also, you can take the bet and never give £2 (although here the monetary position remains unchanged). – hexomino Aug 1 '20 at 11:19
• Yeah, I suppose, but if I were to account for every single scenario, it would be very long, and be very boring. For example, the odds that they kill me after I gave them £2 are very unlikely! – Oliver Aug 1 '20 at 11:41
• I've just realised the third outcome is wrong @hexomino, the statement describes not giving £2, but I've done different sums for some reason. I'll adjust it now – Oliver Aug 1 '20 at 11:45

You should

reject the offer if you prefer having more money to having less money,

because

they can take your £2, not promise you £3 in return, pay you £1 for losing the bet, and end up taking £1 from you overall.

• interesting. but based off the problem statement, if you give £2, the one betting has to receive £3 (profit of £1) as it's a promise. – FishingCode Jul 31 '20 at 9:55
• I have an interesting idea, what if you do not give the 2 pounds? Would it make the statement a vacuous truth? You would still have to give the 1 pound though... – Curiousmind Jul 31 '20 at 10:28
• @FishingCode Nope: the promise itself is subject to the bet. "I bet that if you give me £2 then I promise to give you £3". (In the original wording, which OP has since edited.) It's not "I promise that if you give me £2 then I will give you £3, and I bet £1 that I actually will". Though even in the latter case, the fact that they're proposing the bet seems to indicate that their promise is in some doubt, and I still wouldn't take them up on the offer. – Gareth McCaughan Jul 31 '20 at 11:59
• I think the reason why this answer works becomes crystal clear if you replace £2 and £3 with two higher quantities in the original statement, for example "I bet you £1, that if you give me £100 I will give you £500 in return" – melfnt Jul 31 '20 at 13:29

Should you take this bet?

Yes. But don't give them anything.

Because:

The premise that your bet is founded on is "if you give me £2 I will give you £3 in return". As a logical proposition, this is equivalent to saying that "you give them £2" implies "they will give you £3".

Mathematically speaking, an implication is true whenever either the first proposition is false or the second one is true. (E.g., "If pigs have wings, then I'm a monkey's uncle" is mathematically true because pigs don't have wings, regardless of my personal family connections. There's no way for this statement to be proven false except by producing winged swine.)

So as long as you don't give them any money, then the first clause is false ("you give them £2"), making the entire implication true according to propositional logic. And since that's what you're betting on, you win the bet, and they have to give you £1. So you come out of it £1 richer.

Interesting question for sure...kinda like one of those teasers but here's my approach/answer to it:

The person being challenged here, should accept the offer. As he/she could walk away with a £2 profit if that persons agree to deal/bet with the person who is initializing the bet. Because, when you give £2 and the person promises to give you £3 in return, that is not including the initial bet he/she placed with you (the challenger) so we have to account for the £1 bet to be included in the profit (£3 - £2).

• This is a great answer to the original question which makes use of the 'I promise' part which I edited. – Curiousmind Jul 31 '20 at 10:06
• now it is an open-ended question :) – FishingCode Jul 31 '20 at 10:06
• Wow, but wouldn't you have to pay the 1 pound? You have lost the bet so it would be no profit. – Curiousmind Jul 31 '20 at 10:15
• exactly, right. because in your other answer you had just provided, it would be a loss and not a profit for the bet-initializer. – FishingCode Jul 31 '20 at 10:17
• I don't see how the original wording helps this answer at all. The original wording was nonsensical, but I can't find any way to interpret in a way that makes the person considering the offer gain \$2. If anything, "I bet that I will make a promise to you" is even worse. – Brilliand Jul 31 '20 at 22:06