ThatDonGuy
Joined: Jun 22, 2011
• Posts: 5353
December 3rd, 2019 at 4:03:38 PM permalink
And after only 3 1/2 years, I finally managed to answer the original question:

Given unlimited time and bankroll - without both of which, you have a stop loss condition and the ER is negative for p < 1/2 - the ER for D'Alembert is:

2p + p2 / (1 - 2p)

where p is the probability of winning a single bet
Note that, for p >= 0.5, the value diverges.

For double-zero roulette, p = 9/19, so the ER is 99/19.
Of course, this assumes you can hold out for a loss streak to catch up. With 57 spins/hour, which is fast, a run of 1 billion spins, which is not unusual, will require 2000 years.
unJon
Joined: Jul 1, 2018
• Posts: 3364
December 3rd, 2019 at 5:26:21 PM permalink
Fun thread. Thanks for posting. What does your diverging formula say when p > 0.5 but the wins are less than double (so there’s still a HE). Such as banker bet in baccarat. Does formula not work?
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
ThatDonGuy
Joined: Jun 22, 2011
• Posts: 5353
December 3rd, 2019 at 6:19:51 PM permalink
Quote: unJon

Fun thread. Thanks for posting. What does your diverging formula say when p > 0.5 but the wins are less than double (so there’s still a HE). Such as banker bet in baccarat. Does formula not work?

This particular formula doesn't work, as the amount won no longer equals the number of wins in the run, and can be different depending on the order of the wins and losses.

For example, if X is the amount you make for a win, a run of L L L W W W is -1 - 2 - 3 + 4x + 3x + 2x = 9x - 6, while a run of L L W L W W is -1 - 2 + 3x - 2 + 3 x + 2 x = 8x - 5. The values are the same if X = 1, but otherwise, they are different, and that's just for the two runs of six hands.
ChumpChange
Joined: Jun 15, 2018
• Posts: 2967
December 4th, 2019 at 2:11:23 AM permalink
Keep losing those bets that make up the HA (like Green on roulette or the push on the 12 on the Don't Come), and you'll be less likely to recover to 15 Losses and 16 Wins, so you may just want to reset your D'Alembert when you get back to the plus side by 2-3 wins (+\$60 to +\$100 in this example). I'm having fun playing the Field Bet. I got demolished playing the Don't Pass. I got demolished on the Field Bet in 38 rolls and lost \$1,070 with a \$150 maximum bet. I had a winning streak of \$1,050 in 145 rolls just before this bust. Lost another \$1,000 in 37 rolls. WinCraps is really hating on me again. I should try this when I'm shooting Field numbers at the table. This digital stuff doesn't seem real anymore. I could go from \$10 to \$40 bets in \$2 increments with a \$400 buy-in; or from \$25 to \$100 bets in \$5 increments for a \$1,000 buy-in.

Bet Total Bet
\$10 \$10
\$20 \$30
\$30 \$60
\$40 \$100
\$50 \$150
\$60 \$210
\$70 \$280
\$80 \$360
\$90 \$450
\$100 \$550
\$110 \$660
\$120 \$780
\$130 \$910
\$140 \$1050
\$150 \$1200

Bet Balance
\$150 -\$1200
\$160 -\$1040
\$150 -\$890
\$140 -\$750
\$130 -\$620
\$120 -\$500
\$110 -\$390
\$100 -\$290
\$90 -\$200
\$80 -\$120
\$70 -\$50
\$60 \$10
\$50 \$60
\$40 \$100
\$30 \$130
\$20 \$150
\$10 \$160
Last edited by: ChumpChange on Dec 4, 2019