Let's say someone plays 1 million hands of fpdw perfectly. So they sit down at a full pay machine and they play 100.76. Just for simplicity let's say they don't insert a players card during the million hands so its zero cashback. What are the odds of them losing outright during this million hand period? thanks.

EV is not enough. It depends on the expected frequency of the structured payouts. Because VP payouts are top-heavy on the best hands, there is more variance than simply evaluating a constant EV return for each play.

You could derive it from evaluating each possible finishing hand combination, and the payout for that hand. Since VP involves strategic play there may be several combinations that occur more frequently than others, so this might be a bit of work.

You could also look at each of the dealt hand combinations, then apply the proper strategic play and evaluate all of the subsequent combinations possible, to derive the above set of finishing hands. This is basically how perfect VP strategy and EV is determined.

@PokerHero77--Those solutions sound like a lot of grunt work but interesting work..I might attempt your third paragraph of advice.
What if I used a computer simulation program and ran 100 million hand sims at .76 and just counted how many times the simulation lost outright and then expounded upon that number? Would that be useless and not mathematically close to accurate?

The standard deviation for the optimal strategy of fpdw is 5.181690/hand according to wizard of odds, and the EV is +.00762/hand

The standard deviation for 1,000,000 hands would then be 5.181690*sqrt(1,000,000) = 5181.69 units per million hands, and the EV is 1,000,000*.00762 = 7620 units per million hands.

To break even would be a -7620/5181.69 = -1.47 SD result. By looking up this value in a table the chance of a breakeven result or better is 92.92%

Thanks for the answer browni3141..sounds extremely unlikely to lose outright after a million hands. I'm planning on grinding .25 fpdw for some side income and just wanting to know the worst case scenario so I can figure out how much money I need to set aside for living expenses.

I trust WizardOfOdds did the correct variance/std. dev. calc, from the weighted variances of all the possible outcomes.

So the std. dev. number above is the way to go.

I took these stats off of Dunbar's Risk Analyzer for Video Poker. This is what your Risk of Ruin for quarter FPDW looks like per the size of your bankroll:

$2700............25%
$3100............20%
$3700............15%
$4500............10%
$5800..............5%
$7500..............2%
$9000..............1%

The great thing about FPDW is there is only a 1.01% loss rate between royals. The royal odds are 45,282 and the payoff is 800 for 1 so 800/45,282 = 1.77%. Subtract that from 100.76% and it's a 98.99% return between royals.

The average cost to produce a royal is easy to figure:

45282 X 1.25 X 1.01% = $572 so the average profit per royal is $228.

20 years ago I played FPDW with as much as 1% cashback. That put the cost of producing a royal at just $5.72. And I also played the game with a .5% progressive meter hitting royals as high as $2000.