Playing It Smart: Do casinos make money from players going broke?

A bigger bankroll gives you a better chance of reaching any given profit point. The money lets you ride through setbacks with the hope of upswings. But do casino earnings depend on solid citizens with modest budgets busting out before hitting their win goals?

Consider a simple coin flip game. One flip has two equally likely results heads (H), win $1; tails (T), lose $1. Chances are 50 percent of finishing $1 ahead or $1 behind.

A second try has the same 50-50 chance of H or T, but doubles the number of equally-likely outcomes. Four distinct sequences HH, HT, TH, or TT each has a one out of four or 25 percent chance.

HH makes you $2 richer, TT $2 poorer; chance is 25 percent of either. HT and TH are each break-even; probability is 2 x 25 = 50 percent. Monetary results are summarized in the Two flip table, along with the probability and expected value (amount times probability) at each level. Prospects of breaking even are twice those of being $2 up or down. Not because chances of the results of a flip differ, but owing to the rules governing payouts. The overall expected value, the sum of the components, is zero a consequence of a "fair" game with an unbiased coin.

 
	Two flip
Bankroll	Probability	Expected value
up $2	25%	+$0.50
break even	50%	$0.00
down $2	25%	-$0.50
overall	100%	$0.00

A third flip again doubles the number of equally-likely sets. Possible sequences and their monetary counterparts are: HHH (up $3); HHT, HTH, and THH (up $1); HTT, THT, and TTH (down $1); and TTT (down $3). This boils down to one way of finishing ahead $3, three of being up $1, three of ending down $1, and one of closing $3 behind. Each "way" implies 12.5 percent probability. The Three flip table indicates that chance of profits or losses fall as amounts rise, while overall expected value stays at zero.

Three flip
Bankroll	Probability	Expected value
up $3	12.5%	+$0.375
up $1	37.5%	+$0.375
down $1	37.5%	-$0.375
down $3	12.5%	-$0.375
overall	100.0%	$0.000

Four flips once more doubles the number of equally-likely possibilities, to 16. These are HHHH (up $4, one way) HHHT, HHTH, HTHH, and THHH (up $2, four ways); HHTT, HTHT, HTTH, THHT, TTHH, and THTH (break even, six ways); HTTT, THTT, TTHT, and TTTH (down $2, four ways); and TTTT (down $4, one way). The chance associated with each outcome is one out of 16 or 6.25 percent. The monetary distribution is summarized in the Four flip table. As before, the more extreme the dollar value, the less likely it is. But overall expected value doesn't change.

Four flip
Bankroll	Probability	Expected value
up $4	6.25%	+0.25
up $2	25.00%	+0.50
break even	37.50%	0.00
down $2	25.00%	-0.50
down $4	6.25%	-0.25
overall	100.00%	0.00 

The bane of many players is a finite bankroll. This leads to the "gambler's ruin" inability to recover from a setback by playing on. Some bettors think casinos make more money from players going bust and being unable to continue, than from edge.

Let's see. Say you have only $2. Two tails in a row at the outset (likelihood is 25 percent) spell r-u-i-n. Possible outcomes would therefore be HHHH (up $4, one way); HHHT, HHTH, HTHH, and THHH (up $2, four ways); HHTT, HTHT, HTTH, THTH, and THHT (break even, five ways); TT (down $2, 25 percent), HTTT and THTT (down $2, two ways). The table reveals that gambler's ruin leads to fewer break-evens, more $2 losses, and, of course, you can't lose $4.

Four flip with gambler's ruin at $2 
Bankroll	Probability	Expected value
up $4	6.25%	+$0.25
up $2	25.00%	+$0.50
break even	31.25%	$0.00
down $2	37.50%	-$0.75
down $4	0.00%	$0.00
overall	100.00%	$0.00

The table also shows that your opponent gains no advantage from your limited bankroll because net expected value is still zero. This effect recalls the insight of the inkster, Sumner A Ingmark: