Happy New DECADE, Bjt!!

More to come, I'm still trying to figure out why I can't post more than a few lines at a time.

 

3. Consequently, the long-term probability is that any blackjack player, regardless of his betting method or the rationale behind it, is likely to win--or lose--about the same number of consecutive hands.

In general, the formula is

P(n wins) = 0.478 ^ n
P(n losses) = 0.591 ^ n

If you know anything about exponential decay, then you know that the case of 0.478 decays much more quickly than 0.591 and that the rate of decay accelerates as n increases. We have already seen this in comparing n=2 to n=3 above. The case of n=4 should be of particular interest to BughouseMaster, given our private conversations on this. I'll leave it as an exercise

4. Consequently, if you are betting more money after a winning hand and less money after a losing hand, your average amount wagered during consecutive winning hands will be greater than your average amount wagered during consecutive losing hands and your profits from play will be greater than your losses. It's that simple!

No. No matter how you limit your progression (2, 3, 4 or more stages), and for a streak of any length you choose, you will experience more losing streaks than winning ones and your losing streaks will be of much greater length. Furthermore, your losses will be greater than those of a flat bettor because you will be betting more, in total, during those more frequent losses.

It's that simple.

 

OK, as you can see, I figured out what was causing my posting problems. For those following along, you may want to re-read the entire series of posts, as there have been some edits. Sorry to have ended up posting it all in pieces. If anyone wants it, I can re-post it all as one post.

 

There is one more flaw in Thomason's logic that needs to be exposed. It's a bit subtle, but here it is. He says that the results of your overall play can be partitioned into sets of winning streaks and losing streaks. That part is true. If you look at any pattern of wins and losses, you can separate it into streaks of 1, 2, ..., n wins and losses. I showed above that, for blackjack, his assertion that the numbers of each kind of streak will be roughly equal is grossly false. For blackjack, you will see more losing streaks than winning streaks and the losing streaks will be longer.

But what if the game is not blackjack? What if the game is coin toss? Each bet is made on the toss of a fair coin. In that case, the probabilities of the losing and winning streaks would in fact be exactly equal. You would expect to experience the same number and length of each kind of streak. His theory is that, if you bet more after a win, then your average bet while winning will be higher than your average bet while losing. If the length and frequencies of both kinds of streaks are equal, then you should come out ahead. How can this be? We all know that the house edge on coin toss is zero and most of us know that no betting progression can overcome the house edge.

The flaw in his model is that, when playing the progression, not every win comes on an increased bet and not every increased bet results in a win. In reality, every (positive) progression begins with a small bet, which may win and every progression ends with a loss, which may be on an increased bet. The logician then says to the mathematician that you can safely ignore the results of the first and last bet in the progression, because the longer winning streaks will overcome the effect of those bets. But you can't. For it is in these results that the expected balance is achieved.

Let's look at the possible progression sequences and their individual contributions to the overall result. The overall expected result is the sum of every possible progression result. Thomason talks about positive progressions (i.e. betting more after a win). Here is a chart I made using a spreadsheet for the case where we start with a 1 unit bet, bet 2 units after a win, and return to 1 unit after a loss. The progression of length 1 is a single loss, the progression of length 2 is a win followed by a loss, length of 3 is 2 wins followed by a loss and so on.

In the chart we can see that:

• A progression of length 1 (a single loss) results in a loss of 1 unit and occurs 50% of the time. A progression of length 2 (WL) also loses 1 unit (due to the win of a small bet combined with the loss of an increased bet) and occurs 25% of the time. So right out of the gate this progression loses 1 unit 75% of the time.
• Starting at length of 3 (WWL) the progression starts to make money but only 1 unit (due to the win of the initial small bet and the loss of an increased bet) and this does not make up for the losses of the shorter sequences. Also, the progression of length 3 only occurs 12.5% of the time.
• As the sequences get longer the amount won increases, but the frequency decreases by an amount such that the bigger wins do not occur frequently enough to bring us into profit.
• I have included sequences up to a fairly ridiculous length of 37 (36 wins followed by a loss) to show that at that point, the net return is very close to zero. You can add as many lines as you want and the net return will get closer and closer to zero, but will not quite get there. The full result would include all sequences of length to infinity and the full return converges to zero at infinity.
  
• In this example, we start with a bet of 1 unit, bet 2 units after each win and begin a new progression with 1 unit after a loss. However, whatever you set your bets to for each stage of the progression, the net result will converge to zero, as expected.

I can provide the spreadsheet for anyone who is interested.

Code:

P Win        P Loss        Base Bet        After Win        After Loss
  50%          50%             1               2                1
                             
Progression                             
  Length     Probability       Result        Return              Net
-------------------------------------------------------------------------
    1        50.0000000000%     -1       -0.5000000000      -0.5000000000
    2        25.0000000000%     -1       -0.2500000000      -0.7500000000
    3        12.5000000000%      1        0.1250000000      -0.6250000000
    4        6.2500000000%       3        0.1875000000      -0.4375000000
    5        3.1250000000%       5        0.1562500000      -0.2812500000
    6        1.5625000000%       7        0.1093750000      -0.1718750000
    7        0.7812500000%       9        0.0703125000      -0.1015625000
    8        0.3906250000%       11       0.0429687500      -0.0585937500
    9        0.1953125000%       13       0.0253906250      -0.0332031250
   10        0.0976562500%       15       0.0146484375      -0.0185546875
   11        0.0488281250%       17       0.0083007813      -0.0102539063
   12        0.0244140625%       19       0.0046386719      -0.0056152344
   13        0.0122070313%       21       0.0025634766      -0.0030517578
   14        0.0061035156%       23       0.0014038086      -0.0016479492
   15        0.0030517578%       25       0.0007629395      -0.0008850098
   16        0.0015258789%       27       0.0004119873      -0.0004730225
   17        0.0007629395%       29       0.0002212524      -0.0002517700
   18        0.0003814697%       31       0.0001182556      -0.0001335144
   19        0.0001907349%       33       0.0000629425      -0.0000705719
   20        0.0000953674%       35       0.0000333786      -0.0000371933
   21        0.0000476837%       37       0.0000176430      -0.0000195503
   22        0.0000238419%       39       0.0000092983      -0.0000102520
   23        0.0000119209%       41       0.0000048876      -0.0000053644
   24        0.0000059605%       43       0.0000025630      -0.0000028014
   25        0.0000029802%       45       0.0000013411      -0.0000014603
   26        0.0000014901%       47       0.0000007004      -0.0000007600
   27        0.0000007451%       49       0.0000003651      -0.0000003949
   28        0.0000003725%       51       0.0000001900      -0.0000002049
   29        0.0000001863%       53       0.0000000987      -0.0000001062
   30        0.0000000931%       55       0.0000000512      -0.0000000549
   31        0.0000000466%       57       0.0000000265      -0.0000000284
   32        0.0000000233%       59       0.0000000137      -0.0000000147
   33        0.0000000116%       61       0.0000000071      -0.0000000076
   34        0.0000000058%       63       0.0000000037      -0.0000000039
   35        0.0000000029%       65       0.0000000019      -0.0000000020
   36        0.0000000015%       67       0.0000000010      -0.0000000010
   37        0.0000000007%       69       0.0000000005      -0.0000000005

 

The problem with your analyses, Gronbog, is that the progression unit increase amount is wrong . His progression starts with 2 units, not 1, and goes: 2-3-4-5 staying at that 5-unit 4th win until a loss. Therefore, if you start it @ 1 unit, you would only increase to 1.5, then 2, and finally 2.5 units; so the way you have it is losing too much since his progression doesn't dictate a constant increase after the 4th win but rather remains at that same unit amount.

 

You can adjust the progression any way you want to and then repeat the exercise. The conclusion will be the same.

 

I believe the math is right and the house advantage can not be overcome with any progression. Even so I am a progression player and love the big wins occasionally.
My system only stops at table limits. Have been doing for years and probably a loser, but lots of fun.
Tournament play diff, go by Smith and Wong. Had fun at Palace yesterday 3 of our group got wild card and Congrats to AZSKY for making final table

 

AZSKY sure doesn't like posting very much, does he?

 

I have my own ways to erase the house advantage, which is 1 reason why I'm up a shit ton lifetime at this game!

 

Probably up?

 

Dont tell me you're a MARTINGALE player..... cuz if so that's just an exercise in futility!

However, I too, love the HUGE wins I've had with the Thomason progression as well and would never bother counting (esp. how much I'm up lifetime).... I want to have FUN when I play blackjack, eat free meals, stay in nice hotels (have stayed in every major Vegas strip hotel multiple times), win big in BJ tourneys, etc. and counting is NONE of that.... it's just like going to WORK on your off days when you could very easily get that same pay-day by playing the way that a robot can already!