Random Pai Gow Tiles Figures

This thread is about some random and useless info about Pai Gow Tiles that I have discovered.

First, nearly 2/3 of all hands have a predetermined outcome. That is, as long as you choose a viable way to play (for example, with 7-8-10-11 you choose 7/9 or 8/8 and not 1/5), you'll always have the same outcome no matter which way you choose. Here are those statistics:

If the dealer is banking:
- 7.1642% of all outcomes will always result in you winning no matter how you set your hand
- 21.354% of all outcomes will always result in the dealer winning no matter how you set your hand
- 35.5766% of all outcomes will always result in a push no matter how you set your hand
- The remaining 35.9052% of outcomes are the ones affected by your strategy

If you are banking:
- 7.4421% of all outcomes will always result in you winning no matter how you set your hand
- 20.7546% of all outcomes will always result in the dealer winning no matter how you set your hand
- 35.8653% of all outcomes will always result in a push no matter how you set your hand
- The remaining 35.9381% of outcomes are the ones affected by your strategy

The next statistic is a fun one, which is the number of unique strategies. Of the 3,620 unique 4-tile hands:

- 1,922 have only one way to play
- 1,057 have two ways to play
- 641 have three ways to play

This means that there are 1¹⁹²² × 2¹⁰⁵⁷ × 3⁶⁴¹ = 1,055,430,625,879,613,176,101,466,
267,616,523,314,477,729,388,855,722,069,147,250,898,961,429,450,133,703,152,091,
627,719,772,823,796,609,413,664,612,892,541,513,543,706,547,148,484,341,285,043,
215,783,432,720,654,161,095,983,992,005,657,737,190,595,061,431,238,965,268,828,
054,200,067,596,155,022,143,209,072,650,870,771,415,582,836,795,139,576,617,469,
136,268,384,971,028,933,357,308,292,449,394,007,352,043,391,970,147,135,686,113,
937,272,759,090,779,001,749,329,180,775,374,611,456,294,430,505,109,883,195,728,
750,596,032,678,844,183,739,240,145,986,314,420,753,102,499,827,219,997,565,278,
864,017,557,609,406,822,726,414,512,311,052,815,725,358,130,491,645,023,890,006,
232,780,822,973,881,628,753,792,812,384,683,367,636,837,205,342,349,721,613,705,
535,971,296,474,881,338,292,290,520,520,973,921,728,972,582,871,486,395,580,416 unique strategies.

Yep, and after all that, dumbbutts like me will still fail to split a mixed pair and set the hand wrong.

Thanks!

Very interesting stats! I hope my pai gow ability improves to the point where I can discern when my hand is one of the 35% that matters or mattered!
-G

Quote: smoothgrh

Very interesting stats! I hope my pai gow ability improves to the point where I can discern when my hand is one of the 35% that matters or mattered!
-G

It's not 35% of all hands, it is 35% of all outcomes, meaning 35% of all 8-tile combinations of your 4 tiles against the dealer's 4 tiles. So you can't know if it's one of the 35% until after the game is already over.

I did realize that the 35% percentage based was the outcome, so I lazily tacked on the "or mattered." In my first and only pai gow session, the dealers cleared the tiles relatively quickly so it was difficult for me to compare hands -- especially when the point values were tied!