Gambler's Ruin Problem Part 3: Odds in Blackjack

Click here to see part 1 of this four week series.
Click here to see part 2 of this four week series.


We spent the last two weeks talking about the Gambler's Ruin Problem and how it is a gateway to analyze different games like craps and roulette. Here is a refresher of what we have discovered so far, as well as what the Gambler's Ruin Problem is:

Suppose that you find yourself in a city with a casino and you have $60 in your pocket. There is a concert in town that you really want to see, but the tickets cost $100. You decide that you will place $1 bets until you either reach $100 or go broke. Which casino game should you play? How likely are you to reach your goal of $100?

And here were the probabilities we found for all bets in roulette, the pass line bet in craps, and the other two control statistics.

Game	Avg Gain Per $1 Bet	P of Reaching $100
Favorable Game (51-49 odds)	2¢	92.6%
Fair Game (50-50 odds)	0¢	60%
Craps	-1.4¢	28%
Roulette	-5.3¢	1.3%

We found that roulette is a game of minimal strategy; each bet had the same expected value. Craps is a more difficult game because of the many different bet options each with different perks and odds, but players still don't have to worry about utilizing strategies on the fly. Blackjack starts to get into some more complicated strategy building.

Though there are other players at the table, like craps and roulette, blackjack is a game of you against the house (which is represented by the dealer). You will receive two face up cards and the dealer will receive one face up and one face down card. The sum of the numbers on the card is what your total is (J = 10, Q = 10, K = 10, A = 1 or 11). For instance, K 2 would be 12, 9 7 would be 16, A 5 would be 6 or 16. Hands with an ace counting for eleven are called soft hands because they are easier to work with; they can be lowered by ten at any time if need be.

Once you get your two hands, the dealer will ask if you want to "hit" or "stand." Hitting is when you take another card and add it to your total. Standing is when you do not take any more cards and your hand becomes final. If you hit and your total goes over 21, you "bust" and lose your bet.

Once everyone at the table stands or busts, the dealer then turns over his facedown card. He then must hit until his total is 17 or above. Once he gets to there, he must stand. You then lose your bet if his/her total beats yours and you get even money (1:1) if you beat his/her total.

When you get 21 with just two cards (A K, A J, etc), your hand is called blackjack. As you can see on the table above, blackjack pays 3:2 instead of just 1:1. This is one of the perks that you have as a player; the dealer cannot request that you pay him extra because he got blackjack.

As a player, you also have a few additional betting options that I will explain below.

Doubling Down is when you double your bet on the table and agree to only take one more card. With certain types of starting hands, you can increase your payoff in this way.

Splitting is only allowed when you have two cards of the same number. You can "split" the cards, put a second equal bet on the other card, and play both hands simultaneously against the dealer. Most casinos let you re-split cards, but rarely do they let you double down after splitting.

Taking Insurance is only allowed when the dealer's face up card is an ace. You can bet up to half of your original bet that the dealer's facedown card is a 10, J, Q, or K. This bet pays 2:1, as depicted on the table above.

Blackjack doesn't have to deal with as many weird ratios and random bets as craps does, but developing strategy becomes more difficult because of this. Aside from taking insurance, it is all just inflicting different probabilities into a lone 1:1 bet. But how can we optimize our odds on this 1:1 bet?

When you think about it, it seems like you have a pretty big advantage over the dealer. You can stand before 17, hit after 17, split, double down, take insurance, and get paid 3:2 on blackjack. How do the casinos make money? Well, there is one thing that the dealer has an advantage on, which doesn't even cross our minds. Every time you bust, you lose. Even if the dealer busts. This one advantage is what makes blackjack worth it for the casinos.

This doesn't mean that we can't try our best though. A beginner might just make random bets based on intuition, but more experienced players are able to get their odds much closer to 1:1. Let's start with looking at when to hit or stand. First, hard hands:

Dealer's Up Card	Hit Until You Reach
7, 8, 9, 10, A	17
4, 5, 6	12
2, 3	13

So let's say your starting hand is 7 2 and the dealer is showing a 3.

7 2
? 3

Your total is nine, and you hit until you reach 13. So you hit.

7 2 2
? 3

Now your total is eleven, so you must hit again.

7 2 2 K
? 3

Now your total is 21, and you obviously would stand here.

7 2 2 K
5 3 J

You win! Great job. That guideline is probably the most basic part of the strategy, and will come into play in almost every round. If you memorize any of the strategies I discuss here, that is the one to remember.

That was for hard hands (without an ace acting as an eleven). What about soft hands?

Dealer's Up Card	Hit Until You Reach
9, 10, A	Soft 19
8 or below	Soft 18

Let's try it.

A 5
? 9

Our strategy says we should hit.

A 5 7
? 9

Now we have a hard hand of thirteen, so we must switch back to the hard hand strategy and hit until we reach seventeen.

A 5 7 4
? 9

This is one of the most debated parts of blackjack. We have a hand totaling sixteen and we need to hit or stand. Many players are conservative and choose to stand here. However, we know that probability tells us to hit. You could map out all of the possible outcomes of this hand for yourself and the dealer and compare them, and you would find that hitting actually does end up giving you more success.

A 5 7 4 Q
? 9

And you busted.

A 5 7 4 Q
K 9

But you would have lost anyways, so hitting was worth a try on that one. Now let's look at double down and splitting strategies.

Your First Two Cards	Double if Dealer Has
Total 11	10 or below
Total 10	9 or below
Total 9	4, 5, 6
A2 through A7	4, 5, 6

When do you split	If Dealer Has
A, 8	any card
4, 5, 10	never
2, 3, 6, 7	2, 3, 4, 5, 6
9	2, 3, 4, 5, 6, 8, 9

Those are for doubling down and splitting. What about taking insurance? This is the easiest rule of them all.

Don't take insurance

Ever. Let's look at why. The dealer's face down card could be any of the following:

A  2  3  4  5  6  7  8  9  10  J  Q  K

There are nine cards that don't win you the bet and four cards that do. So, the fair odds for the casino to offer would be 9:4. But, they only offer 2:1 or 8:4, meaning that insurance is not worth it (unless you are a card counter).

If you figure out the expected value of your $1 bet incorporating all of these strategies, you end up getting an average loss of 0.5¢. This is much better than craps and roulette! These are the best odds we have seen so far!

Game	Avg Gain Per $1 Bet	P of Reaching $100
Favorable Game (51-49 odds)	2¢	92.6%
Fair Game (50-50 odds)	0¢	60%
Blackjack	-0.5¢	47.8%
Craps	-1.4¢	28%
Roulette	-5.3¢	1.3%

These odds are not bad. In fact, we are not far from having a 50-50 shot of turning our $60 into $100!

This strategy is great. But what I always wonder is why it works. Why do we hit on sixteen if the dealer has a seven showing? Let's figure it out.

First, we determine the dealer's odds of busting with a seven. To do this, we start with the odds of busting with sixteen (the dealer won't bust if they get to seventeen because they are required to stand). This is a sum of their odds of getting a 6, 7, 8, 9, or 10 as their next card value (remember that 10 can be achieved four ways).

P(bust with sixteen) = P(6) + P(7) + P(8) + P(9) + P(10)
P(bust with sixteen) = 1/13 + 1/13 + 1/13 + 1/13 + 4/13
P(bust with sixteen) = 8/13 ≈ 0.615

Now, we can determine the odds of busting with a fifteen. They could either hit and get a 7, 8, 9, 10, or an ace followed by a bust.

P(bust with fifteen) = P(7) + P(8) + P(9) + P(10) + [P(A) • P(bust with sixteen)]
P(bust with fifteen) = 1/13 + 1/13 + 1/13 + 4/13 + [1/13 • 8/13]
P(bust with fifteen) = 99/169 ≈ 0.586

We could then figure it out for busting with a fourteen. They could either hit and get a 8, 9, 10, ace-bust, or two-bust.

P(bust with fourteen) = P(8) + P(9) + P(10) + [P(A) • P(bust with fifteen)] + [P(2) • P(bust with sixteen)]
P(bust with fourteen) = 1/13 + 1/13 + 4/13 + [1/13 • 99/169] + [1/13 • 8/13]
P(bust with fourteen) = 1217/2197 ≈ 0.554

This process can be continued until you get to the probability of busting with a seven, which ends up being around 0.262. All of the probabilities of the dealer's outcomes with certain face up cards are on the table below:

So the chances of our sixteen beating the dealer's seven if we stand is 0.262; the only way we would win is if the dealer busts. If we hit, then we have a good chance of busting but also a chance of getting a number between 17 and 21, each with their own chance of winning.

P(win with an A) = P(dealer busts) + ½P(dealer gets 17)
P(win with an A) = 0.262 + ½(0.369)
P(win with an A) = 0.447

P(win with a 2) = P(dealer busts) + P(dealer gets 17) + ½P(dealer gets 18)
P(win with a 2) = 0.262 + 0.369 + ½(0.138)
P(win with a 2) = 0.730

P(win with a 3) = P(bust) + P(17) + P(18) + ½P(19)
P(win with a 3) = 0.262 + 0.369 + 0.138 + ½(0.079)
P(win with a 3) = 0.809

P(win with a 4) = P(bust) + P(17) + P(18) + P(19) + ½P(20)
P(win with a 4) = 0.262 + 0.369 + 0.138 + 0.079 + ½(0.079)
P(win with a 4) = 0.886

P(win with a 5) = P(bust) + P(17) + P(18) + P(19) + P(20) + ½P(21)
P(win with a 5) = 0.262 + 0.369 + 0.138 + 0.079 + 0.079 + ½(0.074)
P(win with a 5) = 0.963

P(win with a 6, 7, 8, 9, 10, J, Q, K) = 0

Now that we have all of this, let's do an expected value equation to see what our average odds are.

EV(hit on 16) = (1/13)(0.447) + (1/13)(0.730) + (1/13)(0.809) + (1/13)(0.886) + (1/13)(0.963) + (8/13)(0) = 0.295

So hitting on 16 gives 0.295 odds while standing gives 0.262 odds. And surprisingly enough, the hitting odds do end up better. Yes you are likely to bust when you hit, but it is better than banking on the worse odds of the dealer busting. The strategies for all of the other things can be derived in similar ways.

Next week, we will complete our series on the Gambler's Ruin Problem and see if video poker can offer odds as good as blackjack.