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Let's say that I start with \$200 and need to obtain \$500 by betting on an unfair coin. I can bet \$X and will receive \$X more dollars with probability $p$ ($p$ < $\frac{1}{2}$) and will lose \$X dollars otherwise. What is the optimal strategy for achieving \$500? In other words, how would you maximize the probability of attaining \$500?

I'm really not even sure how to start for this question...

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    Intuitively, given that the gamble has negative expectation, I suspect the solution will involve taking the maximum possible bet at each toss. Spreading your bets across more turns just brings your average winnings closer to the expected value, so you want to minimise the number of turns. However, I'll need to think more carefully about whether this intuition can be formally justified... – 2017-02-22
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    This is a question from this year's SPARC summer camp [application](https://docs.google.com/forms/d/e/1FAIpQLSfdyy88b3SKls9r6abmn95_lNxUa41KyqFLsDyuFZoRP-7xJg/viewform?c=0&w=1) and should be put on hold until the due date has passed. – 2017-02-22
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    Question is from the [SPARC 2017 1st Round Application](https://goo.gl/forms/iKoCXB7MdyM5Xu4x1) (Trapped in a Casino question). The application deadline is 1 March 2017, and this question will remain locked as per our [Contest Question policy](http://meta.math.stackexchange.com/q/16774) until after this date. (Deadline has been extended to 8 March 2017.) – 2017-02-22

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