DorothyGale
DorothyGale
Joined: Nov 23, 2009
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August 11th, 2010 at 3:01:05 PM permalink
One of the issues that players seem to most misunderstand in blackjack are "streaks." The feeling is that the player and dealer should get the same number, more or less. That's just not the case. The dealer is much more likely than the player to get long streaks of wins. This is offset by the player being able to double down, split, and getting paid 3:2 on blackjacks, as well as standing on hard hands.

In the following, I considered a 6 deck game. A "win" counts as any amount won by the player > 0. A "win" for the dealer counts as any net loss by the player. This definition is needed because of complicated things that can happen after a split. And so, I analyzed the probabilities of streak lengths.

A streak consists of a losing event, followed by a series of winning events, followed by a terminating losing event. Pushes don't matter and aren't counted. Thus if the player sees LWWPWWPWL, this counts as a streak of length 5 for the player. If the player sees LL then there is no player streak, the LL will be part of a dealer streak. A streak starts with a winning event. The same definition is used for the dealer's streaks.

Here are the probabilities for various streak lengths for the player and dealer:
Streak Length p Player p Dealer
1 0.1333994 0.1152722
2 0.0618375 0.0618375
3 0.0286649 0.0331726
4 0.0132877 0.0177954
5 0.0061595 0.0095463
6 0.0028553 0.0051211
7 0.0013236 0.0027472
8 0.0006135 0.0014737
9 0.0002844 0.0007906
10 0.0001318 0.0004241
11 0.0000611 0.0002275
12 0.0000283 0.0001220
13 0.0000131 0.0000655
14 0.0000061 0.0000351
15 0.0000028 0.0000188
16 0.0000013 0.0000101
17 0.0000006 0.0000054
18 0.0000003 0.0000029
19 0.0000001 0.0000016
20 0.0000001 0.0000008


Here's what you see in 1M hands --

Streak Length # Player # Dealer
1 133399.4 115272.2
2 61837.5 61837.5
3 28664.9 33172.6
4 13287.7 17795.4
5 6159.5 9546.3
6 2855.3 5121.1
7 1323.6 2747.2
8 613.5 1473.7
9 284.4 790.6
10 131.8 424.1
11 61.1 227.5
12 28.3 122.0
13 13.1 65.5
14 6.1 35.1
15 2.8 18.8
16 1.3 10.1
17 0.6 5.4
18 0.3 2.9
19 0.1 1.6
20 0.1 0.8


What's intriguing about this is the huge discrepancy between player streaks and dealer streaks. For example, the dealer gets more than 3 times as many streaks of length 10. On the other hand, the player and dealer get exactly the same number of streaks of length 2 ... and as you've no doubt experienced many times, the player gets more streaks of length 1 by far than the dealer.

For math geeks, the dot product of the player probabilities with the streak length column gives 0.463550775. The dot product of the dealer probabilities with the streak length column gives 0.536426511. These two sum to 0.999977 (longer streaks make up the tail of this probability). These two values equal to the probability of a player win or dealer win, ignoring pushes.

--Dorothy
"Who would have thought a good little girl like you could destroy my beautiful wickedness!"
pacomartin
pacomartin
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August 11th, 2010 at 4:43:28 PM permalink
Thank you dorothy.
It really shouldn't come as such a shock since the dealer has a strong advantage at winning each hand. It is the player's ability to double the bet (including splits) that makes up much of the difference.

Another way to phrase this question is what is the probability that the dealer will have a streak of a given length (say 6 wins in a row), out of a limited number of hands (say 250 which is about a 4 hour session). Compare that number to the possibility of the player having 6 wins in a row. People are surprised at how likely that it is that the dealer will have a killer streak in any given session.
7craps
7craps
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August 12th, 2010 at 3:27:18 PM permalink
Quote: DorothyGale


On the other hand, the player and dealer get exactly the same number of streaks of length 2


This is to be expected when the value of "p" equals 0.5

In the study of streaks, values of "p" between 0.45 and 0.55 will produce very close expected numbers for a streak length of 2.

I have seen the same with the pass and don't pass line bet and even the field bet in Craps also the Banker and Player bets in Baccarat.
winsome johnny (not Win some johnny)
DorothyGale
DorothyGale
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August 13th, 2010 at 7:05:00 AM permalink
Quote: 7craps

250 hands:
dealer streak of 6 wins in a row or more 98.97% / (using .5364 win prob)
player streak of 6 wins in a row or more 74.69% / (using .4635 win prob)



Are you including pushed hands in these 250? It seems you are not.

--Dorothy
"Who would have thought a good little girl like you could destroy my beautiful wickedness!"
7craps
7craps
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August 13th, 2010 at 11:00:01 AM permalink
Quote: DorothyGale

Are you including pushed hands in these 250? It seems you are not.

--Dorothy

update
you are correct.

I was not. I was using your empirical data instead of theoretical.

Here is a photo for the graphs of player and dealer streaks.
win probabilities used
player 0.47
dealer 0.53
Example:
5th solid line from left (streak length of 6 or more, 6+) is the players win streak curve.
Around 130 hands it crosses the 50% probability line.

5th dashed line from left (streak length of 6 or more, 6+) is the dealers win streak curve.
Around 55 hands it crosses the 50% probability line.


graph produced from Excel file using the Wizards formula found in this threads 7th post

Ask the Wizard correction
winsome johnny (not Win some johnny)
7craps
7craps
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January 10th, 2012 at 11:27:05 AM permalink
Quote: DorothyGale

Quote: 7craps

250 hands:
dealer streak of 6 wins in a row or more 98.97% / (using .5364 win prob)
player streak of 6 wins in a row or more 74.69% / (using .4635 win prob)

Are you including pushed hands in these 250? It seems you are not.

--Dorothy

An update to DorothyGale thread
A few more pics.
Thanks to BruceZ over at 2+2 forum the world can have an Excel worksheet that uses VBA code to calculate the probabilities of streaks and multiple streaks for N trials. (SallyMustang asked about this in the thread and BruceZ came through.)
Excellent work.
To read more:
2+2 thread here
streak calculator here
http://www.pulcinientertainment.com/info/Streak-Calculator-enter.html

Dealer win streaks of 6 or more in 250 hands.
~94% chance of at least 1 streak of length 6 or more
~75% chance of at least 2 streaks of length 6 or more
~48% chance of at least 3 streaks of length 6 or more


Player win streaks of 6 or more in 250 hands.


Dealer win streaks
Length of the Longest Run in 250 hands
mean = 7.923434355
Longest RunProbability1 inor less
0 0#DIV/0!0
1 0#DIV/0!0
2 2.14614E-1146,595,308,264.792.14614E-11
3 1.27396E-0578,495.411.27396E-05
4 0.004153122240.780.004165862
5 0.05944101216.820.063606874
6 0.1799838075.560.243590681
7 0.2371101144.220.480700795
8 0.2013604554.970.68206125
9 0.136076137.350.81813738
10 0.081679712.240.899817081
11 0.04610028221.690.945917363
12 0.02519821939.690.971115582
13 0.01354482173.830.984660403
14 0.00721742138.550.991877823
15 0.00382829261.210.995706113
16 0.002025751493.640.997731864
17 0.001070581934.070.998802445
18 0.0005654091,768.630.999367853
19 0.0002985043,350.040.999666357
20 0.0001575626,346.700.999823919
21 8.31585E-0512,025.240.999907078
22 4.38865E-0522,786.030.999950964
23 2.31599E-0543,178.020.999974124
24 1.22216E-0581,822.140.999986346
25 6.44926E-06155,056.590.999992795
26 3.40315E-06293,845.800.999996198
27 1.79573E-06556,875.340.999997994
28 9.47534E-071,055,371.570.999998942
29 4.99964E-072,000,145.450.999999442
30 2.63799E-073,790,762.010.999999705


Length of the Longest Run in 100 hands
mean = 6.490062753
Longest RunProbability1 inor less
0 1.62101E-336.1689919912E+320
1 4.52158E-1122,116,147,732.554.52158E-11
2 5.91254E-0516,913.215.91254E-05
3 0.01168589285.570.011745018
4 0.10478749.540.116532418
5 0.2249666874.450.341499105
6 0.2368938634.220.578392968
7 0.1757027755.690.754095743
8 0.1096323859.120.863728129
9 0.06276261115.930.92649074
10 0.0344070729.060.960897809
11 0.01845245554.190.979350265
12 0.009785685102.190.98913595
13 0.0051597193.810.994295649
14 0.002712413368.680.997008062
15 0.001423628702.430.99843169
16 0.0007465481,339.500.999178239
17 0.0003912892,555.660.999569527
18 0.0002050194,877.590.999774547
19 0.0001073979,311.290.999881943
20 5.62474E-0517,778.590.999938191
21 2.94538E-0533,951.440.999967645
22 1.54209E-0564,846.860.999983066
23 8.07257E-06123,876.310.999991138
24 4.22516E-06236,677.190.999995363
25 2.21109E-06452,266.460.999997574
26 1.1569E-06864,377.850.999998731
27 6.05221E-071,652,288.320.999999337
28 3.16561E-073,158,950.480.999999653
29 1.65548E-076,040,550.760.999999819
30 8.65587E-0811,552,847.800.999999905
winsome johnny (not Win some johnny)
MGBJK
MGBJK
Joined: Aug 31, 2016
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September 7th, 2016 at 1:11:50 AM permalink
Which formula/s have you used to calculate these?
Quote:

Dealer win streaks of 6 or more in 250 hands.
~94% chance of at least 1 streak of length 6 or more
~75% chance of at least 2 streaks of length 6 or more
~48% chance of at least 3 streaks of length 6 or more


Or can you upload the spreadsheets for both dealer and player?
OnceDear
Administrator
OnceDear
Joined: Jun 1, 2014
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September 7th, 2016 at 5:29:33 AM permalink
Quote: MGBJK

Which formula/s have you used to calculate these?

Quote:

Dealer win streaks of 6 or more in 250 hands.
~94% chance of at least 1 streak of length 6 or more
~75% chance of at least 2 streaks of length 6 or more
~48% chance of at least 3 streaks of length 6 or more


Or can you upload the spreadsheets for both dealer and player?


MG,
You are asking a question in a post that is nearly 5 years old?
Beware. The earth is NOT flat. Hit and run is not a winning strategy: Pressing into trends IS not a winning strategy: Progressives are not a winning strategy: Don't Buy It! .Don't even take it for free.
MGBJK
MGBJK
Joined: Aug 31, 2016
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September 7th, 2016 at 11:05:42 PM permalink
@OnceDear,
Wassup OnceDear?!??!

Yup wasn't here 5 years ago, thought this post was interesting and maybe there was some aficionado like yourself, that could explain to me, the formula used to calculate these percentages,.....:-)
JyBrd0403
JyBrd0403
Joined: Jan 25, 2010
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September 8th, 2016 at 4:27:24 AM permalink
Quote: DorothyGale



What's intriguing about this is the huge discrepancy between player streaks and dealer streaks. For example, the dealer gets more than 3 times as many streaks of length 10. On the other hand, the player and dealer get exactly the same number of streaks of length 2 ... and as you've no doubt experienced many times, the player gets more streaks of length 1 by far than the dealer.

For math geeks, the dot product of the player probabilities with the streak length column gives 0.463550775. The dot product of the dealer probabilities with the streak length column gives 0.536426511. These two sum to 0.999977 (longer streaks make up the tail of this probability). These two values equal to the probability of a player win or dealer win, ignoring pushes.

--Dorothy



I'm interested in whether the Dealer and Player getting the same amount of 2 in a row streaks is bogus or not? Seems like a bogus table to me, with dealer and player both getting the same number of 2 in a rows. Am I wrong?

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