Saki and Probability – Episode 19
Posted by 0rion on August 16th, 2009 - 5:32 pm
Saki’s epic win over Koromo in episode 19 was unquestionably amazing, with lots of lens flare, typhoons back and forth across the mahjong table, fiery Terminator eyeball glare, and LOL MAHJONG IS FUN LOL. It was also completely ridiculous and unrealistic (surprise!).
Obviously that’s par for the course with this series, when we’ve just seen a hanchan played that had multiple haitei raoyue (bottom of the ocean) and rinshan kaihou (drawing your winning tile from the dead wall after a kan) declarations made. I was curious, though, just how unlikely this last game really was.
Warning – lots of mahjong diagrams and math ahead. Read on if you have questions about what exactly happened in that last round of Saki 19, or if you’re curious about just how incredibly unlikely Saki’s win really was. If you don’t care about any of that, go ahead and just go straight to the end for the fanservice pictures.
OK, so first let’s consider Saki’s hand on the final turn of the game, and the sequence of events that leads up to her absolutely ridiculous win over Koromo. You can skip past this and right to the math if you already understand the details of what happened in that last game. This is Saki’s hand on the final turn, right before Koromo discarded:
So she’s in tenpai (needs only one tile to win) for a chinitsu hand (all one-suit), and she’s waiting on either a 1-pin or 4-pin. Basically, if she gets 4-pin she can win like this:
And if she gets a 1-pin, she can win like this:
Koromo conveniently discards the 1-pin, one of Saki’s winning tiles, giving her a mangan winning hand worth 12,000 points. Since the point difference between them is so large, however, Saki doesn’t declare a win; instead she declares a kan (4 of a kind). Now Saki’s hand is like this:
She’s still in tenpai, but now she’s waiting on a 4 or 5-pin to complete her hand. If she gets a 4-pin, she can organize her hand this way, making the pair with 4-pins:
And if she gets a 5-pin, she wins this way, making two sequences and then using the remaining 3-pins for her pair:
Whenever you declare a kan, you need to draw an extra tile, since you are using 4 tiles for a single set instead of the usual 3. Saki draws her replacement tile from the dead wall and it’s a 5-pin, one of her winning tiles! That’s her trademark rinshin kaihou, completing a winning hand with the dead wall draw after a kan.
Now Saki’s hand is:
This is already a winning hand, but it’s still just a chinitsu rinshan kaihou, not enough to force an upset. Saki declares another kan with her 2-pin tiles. That makes her hand now in tenpai for a pair wait on a 4-pin, aiming for a pair of 4s and a 3-4-5 sequence. Amazingly, her next dead wall draw is the 4-pin, another rinshan kaihou winning hand.
Of course, it’s still just a mangan, so Saki continues by declaring a third kan, with the 3s, which means she is again in tenpai, this time on a pair wait for a 5-pin. And unsurprisingly, she again draws the necessary tile from the dead wall, the red 5-pin for the win.
So not only did Saki get all 4 of the tiles she needed to achieve this unthinkable win, she did it with a rinshin kaihou every single step of the way, three times in a row!
OK, now to get into the probability of this crazy stunt – it’s actually a far more complex problem than you might initially assume. If you wanted to be really mathematically correct, calculating the odds of drawing a specific tile during a game isn’t a straight probability, you have to calculate the upper and lower bounds of the probability, since you also have to take into consideration the probability that one of your opponents has already drawn the tile.
And if you wanted to get really crazy, you could weight your probability by estimating the likelihood they would hold that tile, based on strategic evaluation of their discards.
Basically all that to say, I’m going to do some simplification in order to produce an estimate that’s easier to calculate. The simplest way to do that is to assume that none of the opponents are holding the tile being evaluated unless we know for a fact that they are holding it.
To determine the probability of a chain of events like this occurring, first we need to figure out the probability of each individual event in the chain. The first and key step was Koromo discarding the 1-pin that enabled Saki’s little kan rampage. If we just go with the simplistic evaluation that she could have chosen any of her 13 tiles to discard, the probability of that first discard that started the whole chain reaction was:
1/13, or approximately 0.0769
A mahjong set is made up of 136 tiles total – 3 suits of 9 distinct tiles each, plus 7 distinct honor tiles. Of those 136 tiles, 44 of them are visible to Saki at the beginning of this whole exchange.
30 discarded tiles in the pond, 13 in Saki’s hand, and the 1 dora indicator tile. We can also discount the tiles in the other players hands, since we know that in this case they are not holding her winning tiles, which subtracts another 39 tiles from the mix.
136 – 83 tiles = 53 possibilities for the first dead wall draw.
After that, Saki’s first dead wall draw was a 5-pin. We know that Koromo is holding one of them, but the other 3 are unaccounted for, so the chances of drawing one were:
3/53 possible tiles ~= 0.0556
A second kan, and now Saki draws a 4-pin from the deal wall. Saki is holding two 4-pin tiles, and Koromo has a third, so only one remains to draw. The available tile count decreases by two, one for the drawn tile and one for the new dora indicator that gets flipped. Therefore:
1/51 possible tiles ~= 0.0196
Finally, Saki needs another 5-pin to complete her hand. There are still 2 potentially available, so:
2/49 possible tiles ~= 0.0408
Now to calculate the intersection of these probabilities, or the chance that they all occur in sequence. I’ll do it the simplified way and just multiply them:
0.0769 * 0.0566 * 0.0196 * 0.0408 = 0.00000348
or a
1 in 287,356 chance as the approximate lower bound (best case scenario).
TL;DR:
That means that even if you played a full hanchan, or 8 games, every single day of your life, you’d probably still not see a stunt like this in 100 years. Basically you have about the same probability of being struck by lightning within the next year as of seeing something like this happen. And that’s the best case scenario calculation!
Calculating an upper bound is a lot more complicated, but some back of the envelope math suggests that when taking all the different factors into account it could be well over a 1 in two million chance.
In other words, what Saki did was basically the equivalent of winning the lottery. Not that you didn’t already know that. LOL MAHJONG IS FUN!!
This post is full of mahjong and math and makes my brain tired. -_-
Whoa…Math…Cool…
Perhaps you saw those Hong Kong movies with similar plots?
Is Saki really that interesting? I got sick of the Mahjong stuff on episode 1…:P
Bernkastel probably gave her a hand
At first glance it seems that you could arrive at tighter bounds or even an exact solution if you used combinatorics. There are also known distributions for the probability of drawing certain objects with and without replacement.
[...] 0rion 2009-08-17 07:32:10 See the original post: Epic Win Anime Blog » Saki and Probability – Episode 19 [...]
I should have heeded the warning…my head hurts now. -_-
Nice math. I’d observe that Saki needed EXACTLY the red 5-pin to get 13 han for her kazoe yakuman, so the probability is even smaller.
Chinitsu (not concealed): 5 han
Toitoi: 2 han
Sanankou: 2 han
Sankantsu: 2 han
Rinshan kaihou: 1 han
Red dora: 1 han
Mahjong is indeed fun. Saki is pretty unrealistic (or maybe it fits in the supernatural category?), but it’s great for memorizing hands, haha.
Good job at those calculations.
And now if someone bothered to not only work out the probabilities of Saki getting that hand, but also the probability of so many awesome hands taking place so often, we’d go crazy.
But well, seeing it in another light, it would have been boring if for the final match they had just gotten the normal hands you can usually expect in mahjong.
[...] Saki’s 1 against 287,365 chance win against Koromo was awesome, be ready for the Absolute Epic next season when Saki [...]
@ Ray
I certainly think Saki is interesting… it’s my favorite show out of everything that’s airing right now, and that’s saying quite a bit considering the competition.
But then, I really like playing mahjong, so it’s inherently interesting to me. I’m watching the show because of the mahjong, not in spite of it.
@ intro
Yeah, but that would’ve required actual work to figure out, plus hardly anyone would have been able to understand if I posted about that. Not only is this easy, it’s much easier for non-mathies to follow along with.
@ Son Gohan
Interestingly… didn’t Koromo have the red 5-pin in her hand? I’m pretty sure she did. I’ll have to go back and double check that.
@ I-K
Yeah, it’s a lot easier to memorize vocabulary like rinshan kaihou when they do it every other hand, lol. As opposed to real life, where you’ll see it once every never.
@ Sergio
Yeah, trying to really dig into the probability of all the stuff that happened in Saki would probably drive me nuts. But yes, I would have been completely disappointed if there wasn’t something totally ridiculous.
(deleted, don’t make me ban you) – 0rion
Wait! Hold on! Let me explain myself! Although the first part of the comment was a bit over the top, I only did it cause you did the same on Sea Slugs:
http://www.seaslugteam.com/archives/2009/08/17/card-captor-of-the-clow-nintendo-wii-is-sellin-now-release-hoe-sora-no-manimami-01/#comment-77926
Please tell me if that comment was made by you, and if it wasn’t, my apologies! And if it was… I’m still sorry! D:
But yea, if that wasn’t you, I delete it right now, and sorry for the misunderstanding!
1. Yes, that other comment was made by me, and yes your response here was funny.
2. I don’t typically moderate comments here; I do, however, draw the line at non PG-rated commentary, which is why I deleted yours.
3. Check comment IP addresses on your blog if you’re not sure who the commenter is, and heck feel free to delete my comment as well. It’s not like it would offend me.
I wish I could check the IP address, but I can’t (only Kabitzin has those blogging powers. All I could do is create/edit my own posts and… that’s pretty much it…. We’re slaves I tell ya!)
Anyways, I was actually quite surprised by your comment! You don’t seem like the type to just impersonate others (of course, you did put your name/website there, so I was glad it was you and not some random person) So in memory of this event, I shall leave the comment as is. It’s not too often that I see you playing around like that ;3
I’m glad this was all cleared up! Lets vow to never do this again!
I love how the final started making statisticians cry. This final hand is just awesome. :3
@ TheBigN
Hey, now there’s a blogging acquaintance I haven’t heard from in a while! How have you been? I noticed you haven’t blogged in a while, but then I’m hardly in a position to criticize someone for their post frequency.
@ J159
I would hardly call it impersonation. =P
It was clearly intended to be a humorous response to your comment about my Gravatar, nothing more. Sorry it caused a freak out!
Freak out?! No, of course not! Nothing of the sort! I only thought that I perhaps lost your friendship again just like the time I talked like a sailor about Akari (which was one-sided cause I thought I lost your love, but in reality, you loved me just the same! I didn’t want a confusion like that to happen again! ;3… but yea, I did freak out alittle, since my email-client was going haywired! I really thought it was a hacker! You’re a sly one 0rion, I’ll give ya that! =3)
Also, I was really going to have a cute Akari picture, but ohhhhh nooooes! I’m definitely going with what Kabitzin said, and since you only like PG-stuff, you won’t even be able to post it on this site! XD
I think it’s easier to hit the lottery than drawing a hand like that. I doubt a hand like that has EVER been made, at least with lottery, people still do hit the jackpot.
Only someone who is in the same situation as Saki where they absolutely NEED that kind of win on the final hand of the game will they even dream of going for such insane draws. I think most people would just end it when they can even then.
But that’s seriously some impressive maths lol
Dude.
This is one of the many reasons I am so glad that we write things together. This is a perfect example of a post I would like to write but never could due to the fact that my mind becomes engulfed in flames whenever I encounter the most basic of arithmetic.
For example…
I was playing a flash-based mahjong game and was thinking of how utterly improbable the likelihood of Saki’s final hand was and the only thing that came to mind was, “there are probably a great deal of numbers involved with seeking an answer to such a quandary.” After that I stopped thinking about it and went back to re-watching paranoia agent. (As a side note, I cannot recommend this series strongly enough, and would like to do a post on it sometime in the near future)
On another note, I think if I had studied probability in high school with the aide of copious fanservice pictures I would’ve had a greater appreciation for mathematics in general.
Well played mayeng.
I like the stat tear-inducing odds…otherwise calling them haxx0r monsters won’t be very apt.
[...] 19, for her near impossible win in defeating Koromo and the rest to enter the Nationals! There are sites discussing the (im)probability of Saki actually managing to turn her non-winning hand into a big [...]
[...] time. Also, here's someone who sat down and did probability for Saki's Rinshan Kaihou stunts Epic Win Anime Blog Obviously we know it's not even remotely possible to pull off 90% of what's done in the show, but [...]
[...] the rest. That is the time when Saki won during episode 19. She had a hand that is practically impossible to get. Watching her pull a Kan one after the other and finally getting that tile that gives her the hand [...]
This is probably amazingly late (yes I know it’s been half a year since the last comment), so sorry about that (I got here, by Googling “rinshan kaihou” looking for the kanji
), but I noticed a slight flaw in the explanation of Saki’s first hand after Koromo discarded the one pin.
You show 1p1p(1p) 1p2p3p 2p2p2p 3p3p3p 4p4p (the parens indicate the called tile, which you show as a sideways tile) as the winning hand, stating a “mangan” worth 12,000 points. Mangan would have been 8,000 as Saki wasn’t the dealer. But you are correct with this layout as being chinitsu (5han) = mangan.
However, this is really a haneman hand:
1p1p1p (1p)2p3p 2p2p2p 3p3p3p 4p4p
This is chinitsu sanankou (5han + 2 han) which gives the requisite 6-7han for haneman.
Once Saki gets her first rinshan kaihou, all of her hands are chinitsu, rinshan kaihou, tsumo (5+1+1han=haneman) all the way until her last hand.
Like I said, just a minor detail, because your main point stands: the odds are utterly ridiculous to get even two rinshan kaihous in a row, much less three (much less the three others she got earlier). And even one is no mean joke. I think I’ve had the chance at least 30 times with called or drawn kans when I was tenpai, and never hit it.
But like you said… c’mon. THREE haitei raoyue? And Saki had all those rinshans before? If any player did what either Koromo or Saki did as many times as they did it, I’d be asking for a review to make sure they weren’t cheating.
But that’s why it’s fiction.
Still love the series. It got me into playing riichi mahjong, so I guess that’s a good thing.
Ciao!