How much more can casino gamblers bet than they bring?

Win or lose in the end, most casino buffs bet lots more money than they bring to the machines or tables. Few aficionados risk big enough fractions of their stakes, or wager on all-or-nothing propositions with such long odds, that they go kablooie on one pass through their fanny packs by losing every coup. Instead, players feed gains back through the gambling mill.

A common way to reckon how much can be bet on a given budget hinges on what the casino theoretically drains off every try by means of its edge. Say someone starts with $100 and bets $1 per spin at a slot machine with 5 percent house edge (95 percent player return). On this basis, after 100 spins, 0.95 x $100 = $95 would be left. After 95 more rounds, the residual would be 0.95 x $95 = $90.25. Another 90.25 tries later (retaining the fractional amounts for computational purposes), the reserve would fall to 0.95 x $90.25 = $85.7375. Then 0.95 x $85.7375 = $81.45063 ... and so on until not enough remains for another spin. In all, this would yield 2,000 $1 rounds – an average gross wager or “handle,” of $2,000 – on the $100 bankroll.

This average handle can be more readily found as the quotient of the stake divided by the edge. For this example, it’s $100 divided by 5 percent, which is $100/0.05 = $2,000. Were the edge 7.5 percent (92.5 percent return), the average gross wager before the $100 is exhausted would decrease to $100/0.075 = $1,333. Moving in the other direction, a 2.5 percent edge (97.5 percent return) would let a solid citizen bet an average total of $100/0.025= $4,000 on the $100 stake.
Dividing bankroll by edge gives an average that’s reliable over tens or hundreds of thousands of players. The results aren’t especially helpful to predict what will happen to any particular individual, though. They ignore the probabilities and amounts of wins and losses from round to round which influence the likelihood of players getting KO’d before reaching the average, or surviving longer – even winning rather than losing. Achieving the average is pure happenstance. Most people will crash sooner and a few will get further or not fail at all. More, an average shouldn’t be mistaken for a level which half fall below and half finish above. That’s a parameter the math mavens call a “median,”something else entirely.

More useful estimates of the action players can anticipate from their mad money can be obtained utilizing what’s known as a “risk of ruin” analysis. This approach accounts for volatility as well as edge. And it gives results in terms of probabilities of gross wagers exceeding or falling short of designated levels, providing more complete insights into how players might fare in practice.

Pretend the 95 percent return machine has a volatility characterized by a “standard deviation” of 7.50 (representative bankroll jump per spin of $7.50 per dollar bet). Based purely on uniform erosion by the edge, folks would still average $2,000 in $1 bets. Risk of ruin calculations show that players would actually have 17 percent chance of at least 2,000 $1 spins with the other 83 percent probability of not getting this far. Outlook for 4,000 or more such wagers, a $4,000 handle on $100 bankroll, would be about 10 percent with 90 percent falling along the way. On the short side, 27 percent would get at least 1,000 spins with 73 percent running dry first.
Enquiring minds also want to know how much play they can get in games with lower edge and volatility. In blackjack, for instance with strict Basic Strategy and decent rules, edge might be 0.4 percent and standard deviation 1.13. Predicated purely on erosion of fortune by edge, a $200 poke with pre-deal bets of $5 per round yields an average of $200/0.004 = $50,000 – 10,000 rounds. Risk of ruin analysis gives the chance of being in action after making this many bets as 19 percent, with the complementary 81 percent failing sooner. The chance of a $200 bankroll supporting a handle of $100,000 with $5 bets – 20,000 rounds – is 11 percent, with 89 percent biting the dust before getting there. This extended action would be offset by players getting fewer rounds – such as 69 percent probability of busting out before betting $25,000 – 5,000 rounds, and 30 percent scraping the bottom of the barrel before having bet $5,000 – 1,000 rounds.

Comparable computations illustrate the implications of raising bets relative to a bankroll. At blackjack on a $200 stake, betting $10 rather than $5, projections based on edge alone indicate that a player would still get a $50,000 average gross wager but in 5,000 rather than 10,000 rounds. Risk of ruin calculations give the shot at surviving for the 5,000 $10 rounds as only 15 percent, as opposed to 19% in the case of 10,000 $5 rounds; for bets of $25 on this bankroll, the edge-only $50,000 gross wager estimate represents only 2,000 rounds, the corresponding risk of ruin probability of reaching this point is 10.5 percent. All of which goes to show that, except perhaps for patrons of the casino in Lake Woebegone WI, everyone can’t be above average. And, as the punters’ poet, Sumner A Ingmark, dolefully declared:
In a casino, the going is tough,
Just being average is not good enough.