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

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.

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.

Quote:7craps250 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

updateQuote:DorothyGaleAre you including pushed hands in these 250? It seems you are not.

--Dorothy

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

An update to DorothyGale threadQuote:DorothyGaleAre you including pushed hands in these 250? It seems you are not.Quote:7craps250 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)

--Dorothy

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 Run | Probability | 1 in | or less |
---|---|---|---|

0 | 0 | #DIV/0! | 0 |

1 | 0 | #DIV/0! | 0 |

2 | 2.14614E-11 | 46,595,308,264.79 | 2.14614E-11 |

3 | 1.27396E-05 | 78,495.41 | 1.27396E-05 |

4 | 0.004153122 | 240.78 | 0.004165862 |

5 | 0.059441012 | 16.82 | 0.063606874 |

6 | 0.179983807 | 5.56 | 0.243590681 |

7 | 0.237110114 | 4.22 | 0.480700795 |

8 | 0.201360455 | 4.97 | 0.68206125 |

9 | 0.13607613 | 7.35 | 0.81813738 |

10 | 0.0816797 | 12.24 | 0.899817081 |

11 | 0.046100282 | 21.69 | 0.945917363 |

12 | 0.025198219 | 39.69 | 0.971115582 |

13 | 0.013544821 | 73.83 | 0.984660403 |

14 | 0.00721742 | 138.55 | 0.991877823 |

15 | 0.00382829 | 261.21 | 0.995706113 |

16 | 0.002025751 | 493.64 | 0.997731864 |

17 | 0.001070581 | 934.07 | 0.998802445 |

18 | 0.000565409 | 1,768.63 | 0.999367853 |

19 | 0.000298504 | 3,350.04 | 0.999666357 |

20 | 0.000157562 | 6,346.70 | 0.999823919 |

21 | 8.31585E-05 | 12,025.24 | 0.999907078 |

22 | 4.38865E-05 | 22,786.03 | 0.999950964 |

23 | 2.31599E-05 | 43,178.02 | 0.999974124 |

24 | 1.22216E-05 | 81,822.14 | 0.999986346 |

25 | 6.44926E-06 | 155,056.59 | 0.999992795 |

26 | 3.40315E-06 | 293,845.80 | 0.999996198 |

27 | 1.79573E-06 | 556,875.34 | 0.999997994 |

28 | 9.47534E-07 | 1,055,371.57 | 0.999998942 |

29 | 4.99964E-07 | 2,000,145.45 | 0.999999442 |

30 | 2.63799E-07 | 3,790,762.01 | 0.999999705 |

Length of the Longest Run in 100 hands

mean = 6.490062753

Longest Run | Probability | 1 in | or less |
---|---|---|---|

0 | 1.62101E-33 | 6.1689919912E+32 | 0 |

1 | 4.52158E-11 | 22,116,147,732.55 | 4.52158E-11 |

2 | 5.91254E-05 | 16,913.21 | 5.91254E-05 |

3 | 0.011685892 | 85.57 | 0.011745018 |

4 | 0.1047874 | 9.54 | 0.116532418 |

5 | 0.224966687 | 4.45 | 0.341499105 |

6 | 0.236893863 | 4.22 | 0.578392968 |

7 | 0.175702775 | 5.69 | 0.754095743 |

8 | 0.109632385 | 9.12 | 0.863728129 |

9 | 0.062762611 | 15.93 | 0.92649074 |

10 | 0.03440707 | 29.06 | 0.960897809 |

11 | 0.018452455 | 54.19 | 0.979350265 |

12 | 0.009785685 | 102.19 | 0.98913595 |

13 | 0.0051597 | 193.81 | 0.994295649 |

14 | 0.002712413 | 368.68 | 0.997008062 |

15 | 0.001423628 | 702.43 | 0.99843169 |

16 | 0.000746548 | 1,339.50 | 0.999178239 |

17 | 0.000391289 | 2,555.66 | 0.999569527 |

18 | 0.000205019 | 4,877.59 | 0.999774547 |

19 | 0.000107397 | 9,311.29 | 0.999881943 |

20 | 5.62474E-05 | 17,778.59 | 0.999938191 |

21 | 2.94538E-05 | 33,951.44 | 0.999967645 |

22 | 1.54209E-05 | 64,846.86 | 0.999983066 |

23 | 8.07257E-06 | 123,876.31 | 0.999991138 |

24 | 4.22516E-06 | 236,677.19 | 0.999995363 |

25 | 2.21109E-06 | 452,266.46 | 0.999997574 |

26 | 1.1569E-06 | 864,377.85 | 0.999998731 |

27 | 6.05221E-07 | 1,652,288.32 | 0.999999337 |

28 | 3.16561E-07 | 3,158,950.48 | 0.999999653 |

29 | 1.65548E-07 | 6,040,550.76 | 0.999999819 |

30 | 8.65587E-08 | 11,552,847.80 | 0.999999905 |

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?

Quote:MGBJKWhich 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?

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,.....:-)

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?