20 replies to this topic

[TheChad08]

Oops, forgot to put my post in spoilers.

Sorry about that.

As for the calculations, I don't know how to write programs to do that for me as there would be thousands of possibilities.

[superprismatic]

Spoiler for I wrote a simulator and

My 1,000,000,000 game simulation gave the following probabilities of winning:
Seat 1: 0.342568700
Seat 2: 0.318651372
Seat 3: 0.338779928

[CaptainEd]

Would you say the winning seat benefited more from 20/21 or from the other two going bust?

[superprismatic]

Would you say the winning seat benefited more from 20/21 or from the other two going bust?

I don't know, but I can modify my program to count those things. I have an appointment soon.
When I return from that, I'll fix up the program. Perhaps knowing such things may make it easier
to analyse the problem analytically.

[Yoruichi-san]

Spoiler for A good starting point

Admittedly the ace duality is a little annoying, but setting that aside, treating it akin to rolling 13 sided dice (with some repeats) and summing the values on the dice might be helpful.

[superprismatic]

Would you say the winning seat benefited more from 20/21 or from the other two going bust?

Spoiler for I did another simulation for the Captain

Out of 1,000,000,000 simulated games:
Seat 1 won 342,546,409 times, 281,000,589 of which were because seat 1 got 20/21, and 61,545,820 of which were because Seats 2 and 3 both went over 21. In this simulation, Seat 2 won 318,668,459 times and Seat 3 won 338,785,132 times.

[CaptainEd]

Thank you, Superprismatic! (And thank you, Y-san. Such a simple question, with so little intuition available for guessing how it should go, and such a tiny, but repeatable effect. Great fun!)

I'm quite surprised how much more important the ability to hit 20/21 is than the ability to be the survivor. Over 80% of the wins due to 20/21!

And yet, even so, the first position is only very slightly stronger than the other two...
Fascinating!

[brifri238]

Similar to Superprismatic I too ran a simulation (over only a mere 100,000,000 hands) and got similar results with Hand1 winning and Hand 3 in second.(Hand1: 34.257% / Hand3: 33.854% / Hand2: 31.889%). I also ran the simulation for a 'REAL WORLD' execution using one deck of cards without replacement. (Other assumptions the same, and in addition: if a hand goes over 21 no further cards are dealt to that person during that round-[this assumption is irrelevant when using an infinite number of decks]). In this case Hand 3 is the most likely to win! - (Hand 3: 34.0045 %, Hand 1 in a close second with 33.9777 % and Hand 2 with 32.0178%)

Edited by brifri238, 02 November 2012 - 04:18 PM.

[brifri238]

I re-ran both my simulations: scenario 1: with an infinite number of card decks, and the more real world scenario 2: with one deck of cards where cards are dealt without replacement. Both simulations were run over 1000,000,000 hands and the previous results and rankings were confirmed:

Scenario 1: (infinite number of decks): Hand 1 (34.2521%), Hand3 (33.8808%) and Hand 2 (31.8671%)

Scenario 2: (one deck without replacement): Hand 3 (34.0031%), Hand 1(33.9674%) and Hand 2 (32.0294%)

Edited by brifri238, 05 November 2012 - 09:50 AM.

[Yoruichi-san]

Wow, that's pretty interesting, thanks brifri .

If anyone is still attempting the semi-analytical approach, I find that comparing the winning chances of each chair during a round is a good way to do it.