In Mahjong , what is the chance to get a hand that allows a player to finish on their first turn?
I wrote a program to simulate mahjong hand dealings.

It's not verified yet, but so far, the ratio appears to be around 1 in 100,000 games. (The output was the chance that the case happened to at least 1 out of 4 players)

I had the fixed program run for a bit (1.7 billion draws):

3004 mahjong in 1057000000 random draws: 0.00028%, 1 in 351864
4392 mahjong in 1526000000 random draws: 0.00029%, 1 in 347449
4930 mahjong in 1695000000 random draws: 0.00029%, 1 in 343813


At this point, the state space of the random number generator is likely rather close to exhausted, so further research requires a different implementation (this is easy, I just did not bother yet).

If the math of @Jan is correct, it would be pretty easy to replace the RNG with a permutator and just perform a whole-space search to calculate a precise result. The above program calculates around 2m hands per second on my box, so it would take 40-ish hours.



Update: I implemented a proper RNG, optimized the program slightly and let it search through 25 billion random draws:

 737s: 15433 mahjong in  5000m random draws: 0.00031  %, 1 in 323981. 6.78m checks/s
1557s: 31080 mahjong in 10140m random draws: 0.00031  %, 1 in 326254. 6.51m checks/s
2281s: 46008 mahjong in 15000m random draws: 0.0003067%, 1 in 326030. 6.58m checks/s
3116s: 60971 mahjong in 20000m random draws: 0.0003049%, 1 in 328024. 6.42m checks/s
4020s: 75921 mahjong in 25000m random draws: 0.0003037%, 1 in 329289. 6.22m checks/s


The last line is the current state (previous lines for picturing variance).