Assume:

This is more a tutorial write up and less a puzzle. So without further ado here you go.Q: You have a certain amount of money. You are playing a coin tossing game where you know that the probability of you winning in a certain toss is 'p'. You can keep tossing this coin as long as you want. How much of your money should you bet on each toss to maximize your overall returns.A: This problem was first described by J.L. Kelly in 1956 and is known as the "Kelly Criteria". It provides a direct formula for what fraction of your money you should be betting. The interesting thing here is that the Kelly criteria has been used in gambling and investment strategists (including the likes of Warren Buffet). So how does one arrive at that formula? The following is an easy and simple explanation for it.Initial amount of money = CPayout if you win = αLoss if you lose = βProbability of winning = pFraction of money to bet = fA key point to note is that sequence of wins don't matter here. For example if you win in the first throw and lose in the second, your bankroll would proceed as shown in the figure below.Notice, all that happens after a win (or loss) is the appearance of a multiplicative factor of (1 + (1+ α)f) or (1 - (1 - β)f). So the final bankroll after a win-loss scenario and a loss-win scenario is the same:If you were to play the game 'N' times you would expect to win 'pN' times. Thus your final bankroll will beSimplifying it and taking logarithms yieldsIt is intuitive to see that G is the average yield per play. If we maximize G, that 'f' would lead us to the best strategy. To get to that 'f', we first find the derivative w.r.t G and set it to zero. (I leave it to the reader to verify that the second derivative is negative, hence the solution to the first derivative yields a maximum)$$\frac{\partial G}{\partial f} = \frac{p(1+\alpha)}{1 + (1 + \alpha)f} - \frac{(1-p)(1-\beta}{(1 - (1 - \beta)f} = 0$$The above equation on simplification yields the optimal 'f' asThe above result is a more generalized version of the Kelly criteria.In order to test this let us try to generate some simulations around a fixed value of (α,β) and other parameters. The chart below shows the variation of returns using the Kelly criteria vs a simple strategy. The parameters chosen were,Initial bankroll = $100Fixed bet size = $6Win Probability = 55%Returns if winning = 5%Loss if losing = 1%Number of games played = 500The red line is the returns from the Kelly criteria while the blue line is from the simple fixed bet strategy. If you run the simulation over and over again, the charts will look slightly different, and the Kelly criterion strategy would typically tend to be more volatile or even under perform the simple betting strategy. Also notice, the Kelly strategy starts converting much better when it is on a winning streak and likewise loses faster when it is on a losing streak.In the real world (say stock trading or in Poker games) there are a lot more parameters at play that just a win loss strategy. Using purely the Kelly strategy may not be a good idea. This is meant to be a guide on a rational asset allocation strategy. A popular and much used idea is to use what is known as "Fractional Kelly". This approach takes a fraction of what the Kelly criteria suggests.The R code used to generate the chart above is shown below. Due to the stochastic nature, a rerun may not generate the same chart as above.If you are looking to buy some books in probability here are some of the best books to learn the art of Probability