## Recreational Math – Canadian Blackjack (Friday 11/5/09)

Note – This is the first in a series of Recreational articles that will appear each Friday. Some typical topics include sports, music, poker, cooking, …

Several years ago I was in Toronto, Canada and went to a temporary casino that had been set up at a national exhibition. I sat down at a blackjack table and realized that although the rules were the same, the game was quite different. The dealer dealt 1 card face up to each player as well as 1 face up to himself. Then a second card was given face up to each player, but the dealer did not take a second card, instead waiting for all play to finish before taking his second card.

I was sitting at third base, which is the last spot before the dealer. On one hand I had 16 while the dealer had a 10. Basic blackjack strategy states that I am supposed to hit that hand, because if I stand I will only win 21.2% of the hands since that is the probability that a dealer showing a 10 card busts. If I hit, 8/13 of the time I will bust, but the other 5/13 of the time I will make a hand between 17 & 21.

As I sat with my 16 it dawned on me that of the 5 cards that could possibly help me, 4 of those cards gave the dealer a hand of 12-15 which he could then bust from. I decided to hit but announced that I was unsure whether it was the right move, much to the chagrin of my new Canadian friends. I tried to explain my reasoning, but lacked the tools (pencil, paper, calculator, StatCrunch, …) to do an analysis of the situation.

Here goes …

Should a third baseman with a hand of 16 stand against the dealer’s 10 in Canadian Blackjack?

Preliminaries

There are some probabilities that I will need, and will display them here. I will spare you the derivation, but can share if you are interested.

Probabilities for possible outcomes when the dealer starts with a hand of 10

 Outcome 17 18 19 20 21 Bust Probability 11.2% 11.2% 11.2% 11.2% 34.0% 21.2%

Probabilities that a dealer busts with hands from 12 to 16

 Hand 12 13 14 15 16 Probability Bust 48.3% 52.0% 55.4% 58.6% 61.5% Probability 17-21 51.7% 48.0% 44.6% 41.4% 38.5%

The bad news is that 7/13 of the time, it does not matter if you stand or hit. You’re damned if you do and damned if you don’t. Any of the cards (7, 8, 9, 10, J, Q, K) will bust you if you hit, and will give the dealer a made hand if you stand. One point that I want to make clear – if you stand on 16 the only way you can win is if the dealer busts.

The Other 6/13 of the Time

Let’s consider the individual cases in which the next card is an A-6.

Next Card is an Ace

If the next card is an Ace and you stand, the dealer has blackjack. You lose, and so do all the others sitting at the table. You won’t make many friends, but you might get to find out if that Canadian health care system is all it’s cracked up to be.

If you hit you will have 17. You will win if the dealer busts (21.2%), push with the dealer 11.2% of the time, and lose 66.6% of the time.

Next Card is a 2

If the next card is a 2 and you stand, the dealer has 12. You will win if the dealer busts (48.3%), and lose 51.7% of the time when the dealer makes his hand.

If you hit you will have 18. You will win if the dealer busts or makes 17 (32.4%), push with the dealer 11.2% of the time, and lose 55.4% of the time.

Next Card is a 3

If the next card is a 3 and you stand, the dealer has 13. You will win if the dealer busts (52.0%), and lose 48.0% of the time when the dealer makes his hand.

If you hit you will have 19. You will win if the dealer busts or makes 17-18 (43.6%), push with the dealer 11.2% of the time, and lose 44.2% of the time.

Next Card is a 4

If the next card is a 4 and you stand, the dealer has 14. You will win if the dealer busts (55.4%), and lose 44.6% of the time when the dealer makes his hand.

If you hit you will have 20. You will win if the dealer busts or makes 17-19 (54.8%), push with the dealer 34.0% of the time, and lose 11.2% of the time.

Next Card is a 5

If the next card is a 5 and you stand, the dealer has 15. You will win if the dealer busts (58.6%), and lose 41.4% of the time when the dealer makes his hand.

If you hit you will have 21. You will win if the dealer busts or makes 17-20 (88.8%), and push with the dealer 11.2% of the time.

Next Card is a 6

If the next card is a 6 and you hit, you lose. Period. You have 22 and it will not matter what happens to the dealer.

If you stand, the dealer has 16. You will win if the dealer busts (61.5%), and lose 38.5% of the time when the dealer makes his hand.

So, where does that leave us?

Here is a table showing the probabilities of winning and losing for each “next card” and expected value (EV) for standing and hitting. (The expected value is obtained by multiplying the probability of winning by +1, multiplying the probability of losing by -1, and summing these values.)

 Stand Hit Card Win Lose EV Win Lose Push EV A 0 1 -1 .212 .666 .112 -.454 2 .483 .517 -.034 .324 .564 .112 -.240 3 .520 .480 .040 .436 .452 .112 -.016 4 .554 .446 .108 .548 .112 .340 .436 5 .586 .414 .172 .888 0 .112 .888 6 .615 .385 .230 0 1 0 -1 7 0 1 -1 0 1 0 -1 8 0 1 -1 0 1 0 -1 9 0 1 -1 0 1 0 -1 10 0 1 -1 0 1 0 -1 J 0 1 -1 0 1 0 -1 Q 0 1 -1 0 1 0 -1 K 0 1 -1 0 1 0 -1 Total -7.484 -6.386

The expected value for standing is -7.484 over 13 hands. The expected value for hitting is -6.386 over 13 hands. You lose less per hand by hitting than by standing, so hitting 16 versus the dealer’s 10 is the right play.

Now for what may be a surprising result – the expected value for standing in Las Vegas blackjack in the same situation (16 versus dealer’s 10) is

EV = 13 [ (.212)(+1) + (.788)(-1) ] = -7.488

This is only off by 0.004 from my Canadian expected value, and this is due to the rounding of probabilities to 3 decimal places.

So – what do I do if I have 16 versus a 10?

First – have a good cry because you are probably going to lose. Next ask the dealer for a small card and hope for the best.

I am a math instructor at College of the Sequoias in Visalia, CA. If there are topics you’d like me to address in future Recreational math articles, send in your requests through the contact page on my web site. Be sure to check out next Friday’s article – “Are students with neat handwriting better math students?”.  – George