Simulating wealth over time while playing a biased coin toss game

Suppose we have a biased coin toss game that can be played an infinite amount of times, with the following attributes:

• The coin can return either heads or tails on a toss

• The player wins when heads is returned

• The coin is biased, and heads is returned with a probability of 0.6, while tails comes up with a probability of (1-0.6) = 0.4

• If heads shows up and the player wins the turn, they will win the stake that was bet

• (e.g. if you bet 1 dollar, you'll win 2 dollars while if you were to lose you would lose that original dollar).

Given this information:

• Write a function that can simulate n runs of this bet, returning the average payout
• Visualize the average payout across your simulations using a histogram
• Revisit your initial simulation, revising your bet amount (expressed as a function of your wealth) to suggest a roughly optimal allocation
• You can read more about deriving an optimal allocation mathematically here, but in this case just showing the sensitivity through simulation will do