A Probablitiy question involving betting, Coin Flipping, and $50k

Bet $1. If you lose, double your bet to $2, then $4, etc until you win again, then return on $1. As long as you don't throw 15 straight losses, this strategy will gain you $1 every time you win.

Just to be clear, this strategy shifts the probability from even chance of winning or losing the same amount of money to a very small chance to lose a ton of money versus a very large chance to win a small amount of money. There is no magic scheme that suddenly allows you to "beat the house".

Derren Brown did a trick where he tossed 15 consecutive heads in a row and he's got the tape to prove it.

But there's more to it than that, he did toss 15 in a row but it took him a very long time. Every time he got tails he started again, eventually he hit a stretch where he got 15 heads in a row but it took ages.

The expected value for any bet is $0... meaning that on average, you will get $0 (and lose $0) for any bet you make. There is no strategy that will gain you money over time.

Engineers strategy is an interesting strategy, but at most he will earn you $1. At worst he will lose you $32,768. He is giving you 32,767/32,768 of earning a dollar, and 1/32,768 chance of losing $32,768. The expected value is still zero.

In my opinion, the best strategy is just to pocket the $50,000. Why waste all that time flipping a coin to risk $32,768. If you are going to gamble, you really should understand the concept of expected value.