1) Does using the Kelly Criterion assume you will repeat n trials at the same probability/payoff? For example, if I have a game where the payoff and odds change every day, can I just calculate the optimal Kelly bet and use that specific bet for each individual game, and I will still never go broke and achieve the maximum growth rate?

2) What is the formula for what you can expect your bankroll to be after n trials; if you know the payoff, probabilities of winning and losing, and the kelly number?

Thanks!

1. The growth-optimal strategy is to recalculate kelly for each individual wager.

However, kelly-betting doesn't guarantee that you'll never go broke in the real world. IRL there are min wagers and even if there weren't, having only $1 is essentially the same as being broke. Fortunately, the risk of ruin when kelly-betting is low IF your inputs are correct: the chance of being reduced to x% of your bankroll is x%. The bigger risk is that your probability estimates are over-optimistic (ie that you're accidentally betting larger than kelly), which would make your actual RoR higher. In fact it's possible to make all +EV bets and still have a near 100% chance of ruin (as close to it as a finite lifespan allows).

2. Multiply your initial bankroll by (growth factor)^n. For instance if you have a 55% chance of winning a sports bet laying -110, then the kelly fraction is 5.5% and your growth factor is the geometric mean outcome: 1.055^.55 * .945^.45 ≈ 1.004
If you start with $1000, then after 100 of those bets you'll have 1000*1.004^100 = $1490 + change

Thanks this answer is really helpful. Could you look over my calculations I'm missing something here and I can't figure it out...

I have a bet where....

p = 0.4858
q = 0.5142

Payoff (b) is 3:2 = 1.5

So Kelly is... (bp-q)/q = 41.7%.

I want to figure out the growth factor like you did above but Im not getting the expected result... am I using the right values/equation?

(1+(.417*1.5))^.4858 * (1-.417*1)^.5142 = .959

The EV on the bet is positive but this number is below 1, so it must mean that my Kelly value is not correct? I got the formula from Ed Thorpes paper for the Kelly value in an uneven payoff

I used the same formula for your example and it gave me a Kelly of 5.5%

?

Isn't the Kelly fraction given by (bp-q)/b?

Yeah you divide by b not q.
p - q/b = 14.3% for your example

(1+.143*1.5)^.4858 * (1-.143)^.5142 = 1.015170651

Oh and I miscalculated the growth in my example. 1.055 should be (1+.055*10/11), changing the result to 1.001378882, so after 100 bets you'd have about $1148.

Yes there is the error. Thanks alot