Calculation: how often is there no Panda 8 Bonus in a baccarat shoe?

Given a 100 baccarat shoes, and an average of 80 hands per shoe, would anyone have an idea how to calculate, on average, the number of shoes with no Panda 8 Bonus win? (hint: it's not 3.45%) :-)

For an 8-deck shoe, and looking at just the Panda 8 Bonus, the following would apply:

Ways to Win: 172,660,763,262,976 / Win Probability is 0.034543
Was to lose: 4,825,737,512,240,380 / Loss Probability is 0.965457
Total ways 4,998,398,275,503,360 / Total Probability is 1.000000

Any suggestions (or fully worked out math) would be greatly appreciated.

Regards.

Check this out:

http://discountgambling.net/?s=Panda+8

If you assume that the probability of losing is always 0.965457 and there are 80 hands in the shoe, then the probability of all 80 hands losing = 0.965457⁸⁰ = about 6 times in 100 shoes.

I did a simulation of 15 million shoes with full penetration, and got (a) an average of 78.4 hands per shoe, and (b) 1 shoe in 16, or 6.25 shoes out of 100, without a Panda 8 win.

Quote: ThatDonGuy

If you assume that the probability of losing is always 0.965457 and there are 80 hands in the shoe, then the probability of all 80 hands losing = 0.965457⁸⁰ = about 6 times in 100 shoes.

I did a simulation of 15 million shoes with full penetration, and got (a) an average of 78.4 hands per shoe, and (b) 1 shoe in 16, or 6.25 shoes out of 100, without a Panda 8 win.

ThatDonGuy, interesting math. So "how often is there no Lucky Max Low in a baccarat shoe?" In theory, it pays more, so should hit less. It's reported earlier:

https://wizardofvegas.com/forum/gambling/tables/24602-saw-lucky-max-baccarat-at-gold-coast-las-vdgas/

to: ThatDonGuy

Hello, and many thanks for the quick reply. I appreciate your feedback.

May I ask what software you used to run the simulation? And what is the basis of the random seeding for the shoes? ... Asking because, as my screen name implies, I'm a hopeless case of an Engineer :-) (also having an applied math degree - but odds-making is not really my speciality!) and I'm interested in the finer details of all the calculations. Anyway, your method kicks out nearly double the percentage of what I expected to see, given my first rough estimates.

Also, I'd like to politely point out that the probability (0.965457) raised to 78.4 hands/shoe returns a value of 6.35 and not 6.25 as you reported... is that a typo there? Hope so! :-)

Using your suggested calculation, I ran a range of total hands/shoe and this gives a variation from 5.79% (with 81 hands) to 6.91% (76 hands) .... a variance of a full percent, which seems to me, at first blush, to be way too big of a range .... but I haven't done the hard calculations yet ... just some basic preliminary calculations in excel (see more on that below).

Something tells me that there's a bit more complexity here - mostly that in calculating the specific wins/losses, I'd think you'd have to explicitly have simulation software "deal out" hands showing every card value and after hands are resolved, evaluate the following statement:

if ( (PLAYER_WIN == TRUE) && (PL:AYER_TOTAL == 8) && (PLAYER_CARD_COUNT == 3) )
{
PANDA_8_BONUS = TRUE;
}
else
{
PANDA_8_BONUS = FALSE;
}


That being said, because I don't have a sufficiently random number generator, nor do I wish to build/purchase one, I plan to make a first try at simulation using the "1000 computer generated baccarat shoes using eight decks" ( created using a Mersenne Twister random number generator and a Fisher Yates shuffle) that is supplied by the Wizard of Odds web site. I realize that 1000 shoes is not the same as 1000000 shoes... but I need to start someplace :-)

In fact, I've already pulled the shoe file, imported it into excel (which can only hold 723 of the shoes!), and them manually calculated the panda 8 win for a limited number of those shoes... I'll need to write some code to efficiently analyze all the shoes in any decent amount of time.

It will be interesting to see the results. I'll post them here when I'm done.

In the mean-time, I would appreciate any additional thoughts or comments.

Thanks again!

to: UCivan

Hello,

Thanks for your reply

I'm looking for math equations or other methods for calculating the expected frequency and number of Panda 8 Bonus wins in a large number of shoes. I'm not so much into counting the Panda 8 ;-)

Do you know of any web links that might fit here?

Regards,

Quote: ThatDonGuy

If you assume that the probability of losing is always 0.965457 and there are 80 hands in the shoe, then the probability of all 80 hands losing = 0.965457⁸⁰ = about 6 times in 100 shoes.

of course the probability in a played out shoe changes
0.965457 might be a good average probability

Quote: ThatDonGuy

I did a simulation of 15 million shoes with full penetration, and got (a) an average of 78.4 hands per shoe, and (b) 1 shoe in 16, or 6.25 shoes out of 100, without a Panda 8 win.

I also
did a fast 1 million shoe simulation with a 13 card ribbon
and got this distribution for the number of Panda 8 per shoe
the average being 2.9 Pandas
and 81.5 hands per shoe

I also show a 20 card ribbon would produce an average of 80 hands per shoe
if that is important
shoe simulation with a 20 card ribbon

I think a near perfect solution would be to play out (well, in a computer)
all 416! possible shoe arrangements
(416! contains 911 digits)
and count and count
the result would be perfect

of course
many shoe arrangements would never be played out in a real casino
Sally

Quote: RadarEngineer

May I ask what software you used to run the simulation? And what is the basis of the random seeding for the shoes?

Also, I'd like to politely point out that the probability (0.965457) raised to 78.4 hands/shoe returns a value of 6.35 and not 6.25 as you reported... is that a typo there? Hope so! :-)

I wrote a program in Visual C# to do it, and use whatever the .NET Math.Random() RNG is.

And it is 6.35% - it's either a typo or what happens when you trust the buttons on a $5 calculator.

Quote: RadarEngineer

That being said, because I don't have a sufficiently random number generator, nor do I wish to build/purchase one

Here's one that passes the Spectral Tests mentioned in Stanford professor Donald Knuth's The Art of Computer Programming, Volume 2:
the first number can be any integer from 0 to 2³²-1
each successive number is (the previous number * 2650845021 + C) mod 2³²
where C is any odd number (but, of course, it has to be the same number every time)
This generates a "random" order of the integers from 0 to 2³²-1.
Of course, the problem is, once you have generated all 2³² numbers, the sequence repeats itself; when that happens, you could always just use a new value of C and generate the next 2³² numbers in a different order.

Quote: RadarEngineer

...I plan to make a first try at simulation using the "1000 computer generated baccarat shoes using eight decks" ( created using a Mersenne Twister random number generator and a Fisher Yates shuffle) that is supplied by the Wizard of Odds web site. I realize that 1000 shoes is not the same as 1000000 shoes... but I need to start someplace :-)

... I'll need to write some code to efficiently analyze all the shoes in any decent amount of time.

... I'll post them here when I'm done.
!

And, as promised.. I finally had some time today to slam out some code to evaluate the file supplied by the Wiz's Web site... I think we are all on the same page now, as the result I see (for shoes with no panda 8 wins) aligns nicely with the other examples given in this thread.

Here's (below) the result that I have, setting the code to first use all the shoes "as is" (that is, play all the hands that are in each shoe, even if some of the shoes have 87 or 89 hands!), then I limited the shoes to a specified hand count (79 to 81).

Thanks for the previous advice and simulations, folks - it;s not often I see sound advice on mathematics :-)




Results using the file "as is" - hands average ~84 per shoe
------------------------------------------------------------------
------------------------------------------------------------------
Total Baccarat hands: 83640
Total Baccarat Shoes: 1000
Total Panda 8 Bonus Wins: 2928

Max Number of Hands per shoe : 84
Average number of hands per shoe : 83.64000
Number of Shoes without Panda 8 Bonus Win : 55

Actual Percentage of Shoes without Panda 8 Bonus Win.: 0.055000
Expeced Percentage of Shoes without Panda 8 Bonus Win: 0.052852
Delta Actual-to-Expected shoes w/no Panda 8 Bonus Win: 0.002148

------------------------------------------------------------------
------------------------------------------------------------------


Results using the file so that each shoe has 81 hands or less
------------------------------------------------------------------
------------------------------------------------------------------

Total Baccarat hands: 80973
Total Baccarat Shoes: 1000
Total Panda 8 Bonus Wins: 2836

Max Number of Hands per shoe : 81
Average number of hands per shoe : 80.973000
Number of Shoes without Panda 8 Bonus Win : 59.000000

Actual Percentage of Shoes without Panda 8 Bonus Win.: 0.059000
Expeced Percentage of Shoes without Panda 8 Bonus Win: 0.058047
Delta Actual-to-Expected shoes w/no Panda 8 Bonus Win: 0.000953

------------------------------------------------------------------
------------------------------------------------------------------


Results using the file so that each shoe has 80 hands or less
------------------------------------------------------------------
------------------------------------------------------------------

Total Baccarat hands: 79995
Total Baccarat Shoes: 1000
Total Panda 8 Bonus Wins: 2796

Max Number of Hands per shoe : 80
Average number of hands per shoe : 79.995000
Number of Shoes without Panda 8 Bonus Win : 60.000000

Actual Percentage of Shoes without Panda 8 Bonus Win.: 0.060000
Expeced Percentage of Shoes without Panda 8 Bonus Win: 0.060077
Delta Actual-to-Expected shoes w/no Panda 8 Bonus Win: 0.000077

------------------------------------------------------------------
------------------------------------------------------------------


Results using the file so that each shoe has 79 hands or less
------------------------------------------------------------------
------------------------------------------------------------------

Total Baccarat hands: 79000
Total Baccarat Shoes: 1000
Total Panda 8 Bonus Wins: 2766

Max Number of Hands per shoe : 79
Average number of hands per shoe : 79.000000
Number of Shoes without Panda 8 Bonus Win : 60.000000

Actual Percentage of Shoes without Panda 8 Bonus Win.: 0.060000
Expeced Percentage of Shoes without Panda 8 Bonus Win: 0.062216
Delta Actual-to-Expected shoes w/no Panda 8 Bonus Win: 0.002216

------------------------------------------------------------------
------------------------------------------------------------------


Results using the file so that each shoe has 78 hands or less
------------------------------------------------------------------
------------------------------------------------------------------

Total Baccarat hands: 78000
Total Baccarat Shoes: 1000
Total Panda 8 Bonus Wins: 2732

Max Number of Hands per shoe : 78
Average number of hands per shoe : 78.000000
Number of Shoes without Panda 8 Bonus Win : 62.000000

Actual Percentage of Shoes without Panda 8 Bonus Win.: 0.062000
Expeced Percentage of Shoes without Panda 8 Bonus Win: 0.064442
Delta Actual-to-Expected shoes w/no Panda 8 Bonus Win: 0.002442

------------------------------------------------------------------
------------------------------------------------------------------

Are you trying to figure out an edge to the panda 8 bet?

No, not really. The only edge I know of is to count it. I'm just looking for information that's not previously been published. Just playing around and amusing myself.