Among emails I receive, a fairly common question is something like this:
Can you tell me the odds of losing six hands in a row at blackjack?
Sometimes it is 5 hands, sometimes 8, sometimes more. No matter, I cringe whenever I get this question.
In either case, there is nothing wrong with the matter-of-fact question being asked, but it is apparent to the recipient that the question is likely a clue to dangerous thinking, whether it be the Hindenburg or the Martingale progression.
I’ll get back to that in a moment. But first, let me respond to the question, very carefully…
The first thing I want to do is change the wording of the question from “What are the odds” to “What is the probability“. Most people asking this question want an answer that is a single number or percentage, representing how likely the event is to occur. “Odds” means something slightly different, although the two are often used interchangeably. If you want to understand the difference, Google it.
There is a crucial piece of information missing from your question. How do you plan to play?
Let’s say you sit down next to me at a blackjack table, and I look over and tell you: “This is unbelievable! I’m on the most horrible losing streak. My luck is so bad, I bet you $100 that I lose the next six hands in a row.” If you were foolish enough to bet me, I would quickly demonstrate how the probability of losing six in a row can be 100%!
But for the sake of your question, let’s assume the player will use accurate basic strategy on the game. I cannot resist the temptation of pointing out that virtually no player who asks this question will know a completely accurate basic strategy! But I’ll play along anyway.
There is another picky detail to consider. I assume you are not asking for the probability of losing exactly six hands in a row, which implies that the seventh hand cannot be a loss. Instead, you want to know the probability of losing six or more hands in a row. We can easily sidestep that complication, if we simply say that you want to know the probability of losing the next six hands of blackjack.
For a typical shoe game, a basic strategy player will win 43.3% of hands, lose 48.0%, and push the other 8.7%.
If I answer the revised and improved question…
What is the probability of losing the next six hands at blackjack, using basic strategy?
Since each hand has a 48.0% chance of being a loss, the answer is 0.48 ^ 6 = 0.012 = 1.2%.
But wait a minute…
That probably isn’t what you meant either. What if you lose the first hand, push the second hand, and lose the next five hands in a row? You would probably want to call that losing six in a row. We should just ignore pushes altogether, and only count hands that end in a win or a loss. Of non-push hands, players win 47.4% and lose 52.6%.
Finally an answer that may satisfy you:
What is the probability of losing the next six hands at blackjack, using basic strategy, ignoring pushes?
The answer is 0.526 ^ 6 = 0.021 = 2.1%.
That is roughly 1 in 47 attempts.
Just because this is a convenient place to do so, I’ll publish the numbers for other lengths of losing streaks.
|Probability of losing n hands in a row, ignoring pushes.|
When players ask questions about how likely it is to lose hands in a streak, it usually means that they are considering a negative progression betting system that would fail if they encounter a certain number of losing hands in a row. Most commonly, they have rediscovered the infamous Martingale system, where you double up after any losing hands to catch up.
The Martingale is the simplest of the negative progression betting systems. The idea is simple. If you lose one hand, just double your bet on the next hand. Eventually, you will win a hand, and make back all of your losses plus a profit of the initial bet. Indeed, this seems to work very reliably, until it doesn’t. The problem is that most people underestimate how likely those “unlikely” streaks of losses really are. Check the table above. The probabilities get small, but not nearly small enough.
If you think the Martingale is a good idea, you are badly mistaken. Yes, it gives you a high percentage chance of winning a small amount, but it does not take long to run out of luck. For a $10 bettor who wants to be able to handle up to 7 losses in a row, he needs to have a bankroll of $2550, and be willing to bet $1280 on the eighth hand. All this, for a $10 win each time.
And he will run into that fatal eighth loss once every 170 trials on average. You can think of that as making $1700 in profit before you eventually lose $2550. (Yes, I am oversimplifying because doubles and splits make the math very messy. The real numbers still make this a very bad idea.)
On top of that, blackjack is a terrible choice for the Martingale system. With the necessary doubles and splits, sometimes both on the same hand, you will actually need a far larger bankroll to play correct basic strategy. If you choose to ignore basic strategy doubles and splits, you can get by with “just” the $2550, but at a substantially increased loss rate on all your bets.
Any progressive betting system is doomed to fail. Any series of negative expectation bets is guaranteed to create a negative expectation overall. If you are just playing for fun, you will lose less money by flat-betting. If you are playing for profit, you need to find a true way to get an edge, such as counting cards.
Hopefully you are reading this before you learn the hard way. After all, we don’t fly airships full of hydrogen any more either.