Ask the Wizard #205
Tom from Modesto, CA
For the benefit of other readers, the Match the Dealer side bet pays when either of the player’s first two cards match the dealer’s up card. A traditional count is not going to be useful against this side bet. Rather, the odds would swing to the player’s advantage if the distribution of cards by rank were unusually unbalanced. It isn’t going to be practical to keep track of 13 different suits. The Big Book of Blackjack by Arnold Snyder, which I highly recommend, has a short chapter on how to beat a similar bet, the ’Royal Match.’ With only four suits to worry about, this side bet is vulnerable to the method described in that book in a single-deck game.
Doyle from Reno
Is there a number you can pick that will maximize your expected return, if the other player picks randomly? What if the other player has a strategy too?
Andrew from Toronto
I hope you’re happy, I spent all day on the second part, and my answer was still wrong. Lest I deprive my readers of the same joy, I won’t just blurt out the answers here. I broke this into two problems, and posted answers and solutions at mathproblems.info, problems 196 and 197.
If you aren’t counting cards, then it doesn’t matter. If you are, then any burned cards should be added to the number of cards left unplayed in the deck/shoe, when making the true count conversion.
The probability of being on the losing end of KK vs. AA is (combin(4,2)/combin(52,2)) × (combin(4,2)/combin(50,2)) = 0.000022162, for each opponent at the table. That is once every 45,121 hands, so your math was right. The expected number of times that would happen in 400 hands is 400 × 0.000022162 = 0.008865084, per opponent. The following table shows the probability of 3 or more instances of having KK against AA in 400 hands, by the number of opponents.
3+ KK vs AA probability in 400 hands
|1||0.0000001145||1 in 8,734,376|
|2||0.0000009133||1 in 1,094,949|
|3||0.0000030658||1 in 326,182|
|4||0.0000072234||1 in 138,438|
|5||0.0000140202||1 in 71,325|
|6||0.0000240728||1 in 41,541|
|7||0.000037981||1 in 26,329|
|8||0.0000563277||1 in 17,753|
|9||0.0000796798||1 in 12,550|
So, yes, I would say this looks fishy. The fewer the players, the more fishy it looks. I would be interested to know where this game was.
Since I cannot control the statistics, my question involves something that I can control, session length (and bankroll). Since a million or a billion number of hands is composed of so many “sessions” of, for example, 300 − 1,000 hands, does it not make sense to play until you either a) reach a pre-established target win amount, or b) play until you recover from a losing streak and end the session at break even?
One last question, can you recommend a source for a software simulation system that can handle all rule variations, stop/loss provisions, extracting “sessions” of variable lengths, and variable hit/stand strategy based upon the size of the bet. I would love to give my approach a shot at the computer.
Tom from Bowling Green, KY
Thanks. I get questions like this a lot. Usually I delete them, but since you buttered me up so nicely, I’ll answer this time. As I state many times, all over the site, all betting systems are equally worthless. There is no magic quitting point. I am not opposed to any winning or losing marker for quitting, but the expected value is no better or worse than flying by the seat of your pants. I’m told that Casino Vérité is capable of simulating what you ar asking about. Finally, in blackjack, the hit/stand decision should not depend on the size of the bet. The right play for a $1 bet is right for a million dollars.