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# Why Casinos May Earn Less from High than Low Rollers

30 August 2006

Say you enter a casino with \$100. What do the bosses expect you to leave behind? The whole \$100, or less? And, if less, do they consider you a better, worse, or equivalent patron betting \$10 or \$25 per coup? The devil is in the detail of how you gamble.

Say you go for broke, playing until you win \$900 or lose \$100. Many folks believe the casino would figure them for \$100 profit minus a small luck factor, ascribing the effect to house edge.

But, imagine a gamble with no edge. The casino has 90 percent chance of getting your \$100. Your shot at winning \$900 before this occurs is the other 10 percent. The house breaks even.

Here's why. Suppose that 100 people each buy-in at the no-edge game for \$100. They play until they win \$900 or go broke. On the average, 10 will succeed -- winning \$900 each for a total of \$9,000. The other 90 will lose \$100 each for a total of \$9,000. That's \$9,000 in and \$9,000 out. Nothing is left for the casino.

With an edge, the casino gets a share as well. Envision it by pretending the game is blackjack, with everyone following perfect Basic Strategy. The house advantage is half a percent.

When bets are a flat \$10 per round, this edge cuts the chance of winning \$900 before losing \$100 from 10 to under 7 percent. On the average, of 100 players, seven will therefore win \$900 and 93 will lose \$100. Winners take away \$900 x 7 or \$6,300 and losers leave \$100 x 93 or \$9,300. The bosses earn the \$3,000 difference, so each of the 100 bettors looks like \$30 to the casino.

What if everything remains the same except that bets are \$25 per round? The chance of victory becomes 9 percent and that of defeat 91 percent. On the average, \$900 x 9 or \$8,100 then flows out the door while 91 x \$100 or \$9,100 stays; the joint keeps the \$1,000 difference. Each of the 100 \$25 bettors represents a theoretical profit of \$10 to the casino.

The disparity arises because the solid citizens with \$100 stakes need longer either to hit their profit targets or go belly-up betting \$10 than \$25 per round. And, taking longer, the \$10 players will end up betting in more rounds and having a larger gross wager or "handle" on which the edge takes its toll. To explore this counter-intuitive phenomenon, \$10 bettors being worth more to the bosses than their \$25 counterparts, work back through the edge and casino "take" to the handle.

For the house to average \$3,000 at half a percent edge, the gross wager would have to be \$3,000/0.005 or \$600,000. At \$10 per round, this is 60,000 rounds -- an average of 600 rounds and \$6,000 handle for each of the 100 players. For the casino to earn \$1,000 at the same edge, the handle would have to be \$1,000/0.005 or \$200,000. At \$25 per round, this is 8,000 rounds -- an average of 80 rounds and \$2,000 handle for each of the 100 players.

A higher edge with all else being equal would lower the likelihood of winning the \$900 before losing \$100 even further. It would also raise the expected earnings from each player.

Rising from 0.5 to 1 percent edge, prospects of prosperity for \$10 players fall from 7 to 4 percent. At 1 percent, an average of four winners would accordingly grab \$900 x 4 or \$3,600 while 96 losers would kiss \$100 x 96 or \$9,600 goodbye. The casino would earn the \$6,000 difference, \$60 per person. Similarly for the \$25 bettors, probability of success drops from 9 to 8 percent. On the average, eight winners would emerge with \$900 x 8 or \$7,200 while 92 losers would be stripped of \$100 x 92 or \$9,200. The \$2,000 difference would be the casino's commission, \$20 per player.

Alternate exit strategies would yield other results. But the underlying theme would still be that neither bankroll nor bet size alone are good indicators of a player's monetary worth to a casino. They may even lead to pampering the wrong people. As the songster, Sumner A Ingmark, surely suspected when he scribbled:

Conclusions falsely prejudicial,
Arise from data superficial.