I recently learned about a new betting strategy for blackjack, called 1-3-2-6. I’m not a big blackjack player, since my money seems to evaporate so quickly in that game, but I like learning about new gambling strategies.
So for this strategy, unlike the Martingale system, it doesn’t require an infinite bankroll. You progress along a sequence of bets: 1, then 3, then 2, then 6, and you only go to the next number if you win a hand. If at any time you lose a hand, you go back to 1. So let’s say you place your initial bet of 1 unit (something at least the minimum bet, probably $10 unless you’re a big gambler). If you win the hand, you keep your winnings there and place an additional unit to bet, to bet 3 units on the next hand. If you win that one, take your winnings and one unit back from the bet to leave 2 units for the next hand. Now if you win that hand, leave your winnings from that hand on the table and place 2 more units for the bet, for a total of 6 units. Now if you win that hand, the sequence is complete, and you go back to 1. If at any point you lose a hand, you go back to 1. If you push, you leave the bet there.
In the worst case, you lose on the second bet. You would gain 1 unit from the first bet and then lose 3 on the second, leaving you with a loss of 2 units. Even if you did this six times, for a loss of 12 units, you could make it all back by completing the cycle just once, since 1 + 3 + 2 + 6 = 12. But it can’t actually do better than betting the same amount every time, right?
It’s hard to tell at first glance. If we denote a win as W and a loss as L, then we can see LLLL would result in a loss of 4 units under either strategy. WWWW, however, would result in a profit of 4 units under the standard strategy, but 12 units under 1-3-2-6. WLWL would break even under the standard strategy, but would cause a loss of 4 units under 1-3-2-6. WWLW would net you 2 units under the standard strategy, but would get you an extra unit of profit under 1-3-2-6. So there are situations where they’re equally good, and there’s others where one does better than the other, but no clear favorite. That means we have to do some math…
Let’s consider a very simplified model where we assume the probability of winning each hand is p. (This is grossly oversimplified because the probability varies with each hand, since the deck isn’t reshuffled between hands.) And let’s also assume that every time you win, you win 1 unit, and every time you lose, you lose 1 unit. (Again, grossly oversimplified since optimal strategy involves splitting and doubling down when appropriate, not to mention if you hit a blackjack, you get paid more than your bet.) Under a standard strategy where you just bet the same amount on every hand, your expected profit for 4 hands would be 4p + 4(1 – p) = 8p – 4. Under the 1-3-2-6 strategy, you’d have to consider 16 different sequences of wins and losses, since your bet sizing is dependent on past results. I guess I should make a table!
| Sequence | Profit | Probability |
|---|---|---|
| WWWW | 12 | p4 |
| WWWL | 0 | p3(1 – p) |
| WWLW | 3 | |
| WLWW | 2 | |
| LWWW | 5 | |
| WWLL | 1 | p2(1 – p)2 |
| WLWL | -4 | |
| WLLW | -2 | |
| LWWL | 1 | |
| LWLW | -2 | |
| LLWW | 2 | |
| WLLL | -4 | p(1 – p)3 |
| LWLL | -4 | |
| LLWL | -4 | |
| LLLW | -2 | |
| LLLL | -4 | (1 – p)4 |
My table is undoubtedly ugly, since I leave my CSS to the whims of my installed theme. Anyway, to get the expected profit, we can just multiply the profit times the probability for each row, and sum those up. If I did my math right, it should be 12p4 + 8p3 – 8p2 + 14p – 4. If you plug in p = 0.5 in there (totally unrealistic, I know), you get 2.75. So you can expect to make a profit with this strategy, if p is high enough!
OK, not really. In reality, under this strategy, each set of 4 consecutive hands isn’t independent, because of the way it cycles. A lot of the 16 possible four-hand sequences above don’t end right before the beginning of a cycle, so you can’t just do the fairly straightforward probability calculation I did above, unlike for the standard betting scheme, where each hand is independent of any other ones.
I fooled you! I just wanted to make it seem sexier than it actually was. OK but I’m not smart enough to actually figure out the exact expected probability. I kind of just wanted an excuse to make a table to see how it looked. But next time I’m in Vegas, I’m going to try it!