Discount Gambling

My local Barona Casino offers a $1 progressive blackjack side bet that pays when your 20 loses to a dealer 21. The payout depends on the number of dealer cards, where the payout increases with the number of cards in the dealer’s hand. The progressive pays out when your 20 is beaten by a 7-or-more card dealer 21.

*calculated using basic strategy (includes surrenders and other decisions that are not optimal for the sidebet).

**magic card average frequency assumed at 1-every-26 hands, and average 3 cards/hand

So, the break-even point for the jackpot is about $100,000. I used basic simple strategy and a 6 deck shoe in this analysis.

Effect of Shoe Depth on EV

While looking for a counting edge on this side bet, I ran into a very pronounced effect of shoe depth on the bet EV. I was expecting the random distribution of cards at the end of the shoe to distribute the EVs around 0. Well, I did find a roughly bell-shaped distribution of EVs, but I discovered that the average return decreases sharply with the shoe depth.

What this means is that as you reach the end of the shoe, the average return of the bad beat side bet decreases very quickly. Most bets don’t behave like this. You want the bet to have the same average return at the start of the shoe as at the end of the shoe. Imagine if blackjack started at a 0.5% house edge for the first hand, but for the last hand ended up at a 50% house edge, on average. Most people wouldn’t stand for that (if they knew it was happening). [You’d probably sense such a bias on an “even-money” bet, but not for a bet that hits < 2% of the time.]

I've plotted out some curves to shoe the effect of shoe depth on the bad beat EV. I set the jackpot at $100k, so the EV for a full six-deck shoe is near 0 (magic card not included). Then, I plotted EV vs. number of decks in the shoe, assuming no missing cards (i.e., a neutral shoe). So, in the graph below, 0.5 decks means half a deck.

The above graph shows the big disparity between playing the bonus at a 6-deck shoe game vs. a double-deck pitch game. The graph also reflects the bet’s average as a function of decks remaining in the shoe. Taking an extreme example (1/2 deck remaining in shoe; most houses shuffle before this point), the distribution of actual EVs is sampled in the below graph. Note how it’s shaped mostly below the -41.5% average return of the above graph.

This effect is probably due to the fact that you need a lot of perfect cards to make a 7-or-more card dealer 21. When you run low on cards, there’s probably much fewer ways (percentage-wise) to make the perfect hand. Some players might intuitively suspect this, or at least would not be surprised when they see these results.

So be careful with this bet. It gets a lot worse than you think it might, especially at the end of the shoe.