I have a hypothetical question based of the logic of 50/50 gambling.

Question

If you start at a certain bet and every time you lose you double the amount you are betting until you win, would you be profitable (If you had unlimited money)?

To make it more clear if I bet $50 and lose i am negative $50. So the next game I Bet $100. I lose, so I bet $200. I lose the $200, and I am now negative $350. If I win the next bet of $400 I have made $50 profit.

If you can double your bet for an unlimited amount of times until you win, you will always make a profit of the amount of your very first bet.

Are there any fallacies in this logic, assuming there is no house cut to take away from the winning pot?

Answer 1

lol.... if you had unlimited money then yes you have much more than a profit....

 

but anyway yes, there is a profit and it is EXACTLY the amount

of the first bet.

 

Here's the proof.

 

To keep things simple, the proof is based on $1 bet and done by induction.

 

1+2 = 3 = 2^2-1
1+2+4 = 7 = 2^3 - 1
1+2+4+8 = 15 = 2^4-1

in general,
1+2+4+8+.....+2^(k-1) = 2^k - 1  <---- this is the induction hypothesis.

Induction proof shows:
1+2+4+8+.....+2^(k-1) + 2^k =

2^k - 1 + 2^k =  <---- substitutes the induction hypothesis

2*2^k - 1 =  <--- adding the same number to itself is the same as multiplying it by 2

2^(k+1) - 1 ; <--- property of exponents

 

which completes the proof that 1+2+4+...+2^(k) = 2^(k+1)-1

So on the next bet:
2^k - (1+2+4+8+16+.....+2^(k-1)) = <--- subtracts previous losses
2^k - (2^k-1)    <--- from previous theorem

=2^k - 2^k + 1

=1 >0  so yes there is a profit since the result is positive of exactly $1

 

Generally speaking, since you can multiply these equations by

whatever amount you want, you will always come out ahead by the

FIRST amount that is bet, prior to doubling down.

 

Notice in your example, the first bet was $50 and you came out exactly $50 ahead.

 

Suppose it is 100,200,400,800. That's 1500 in losses. You win 1600 on the next which is

$100 ahead

 

The proof I have done for you shows that you will always win back all

of the money and end up with a profit on whatever your first bet was.