Mah Jongg probability

Howdy!

Can someone help me figure out the odds for a mah jongg scenario, which seemingly never happens?

In the game are 2 sets of 4 tiles, called flowers. These are special tiles that if you get all 4 tiles of a set, you get 2 extra points (or "faan") if you have a winning hand.

I have gotten all four tiles once, but have never seen anyone get both sets: all 8 tiles.

There are 144 tiles altogether, and each of the four players takes 13 tiles at the beginning of the deal.

That leaves 92 tiles, so you have a chance to draw up to 23 more tiles. (I say "up to" because during the course of the game, your turn often gets skipped because someone calls "pong" — taking the discard of another player.)

You can never take a flower from another player — you must draw it yourself. When you draw a flower, you set it aside and it's not part of your hand of 13 tiles.

A complicating factor is that when you draw a flower, you immediately draw another tile so that you always have 13 tiles for your hand. In other words, after you draw a flower, it's possible (albeit highly unlikely) to draw 7 consecutive flowers.

But maybe that aspect can be ignored, and let's just say the parameters are: you're trying to get 8 specific tiles from picking 36 tiles out of 144 total tiles. What are the odds?

Is that scenario like playing keno, but only 8 numbers are drawn out of 144 balls, then you pick 36 numbers and hit all 8?

The first could be one of 36 tiles in 144, so the chances are 36/144.
The second : 35/143.
etc.
The eighth : 29/137.
The product of all these are about 8 in a million.

"you're trying to get 8 specific tiles from picking 36 tiles out of 144 total tiles. What are the odds?"

Your question is a bit unclear. If you are picking 36 tiles at random from 144 and you are hoping to get 8 specific tiles, I would say the probability is:

c(8,8)*C(136,28)/C144,36) = 0.00000804 or about 8 in one million.

Same as above post!

Wow, thanks for doing the math. I thought it was more complicated than the “eliminating a card” formula!

So with that probability, if you played a full game of Mah Jongg (16 hands) every day, it takes an average of 21.4 years before you’d see someone get all 8 flower tiles!

Quote: gordonm888

Your question is a bit unclear. If you are picking 36 tiles at random from 144 and you are hoping to get 8 specific tiles, I would say the probability is:

c(8,8)*C(136,28)/C144,36) = 0.00000804 or about 8 in one million.

Same as above post!

Thank you, that’s what I meant. It’s basically getting 8 specific tiles during the course of random picking.

Quote: charliepatrick

The first could be one of 36 tiles in 144, so the chances are 36/144.
The second : 35/143.
etc.
The eighth : 29/137.
The product of all these are about 8 in a million.

So the odds of getting all four tiles of a set are (36/144)*(35/143)*(34/142)*(33/141) or 3 in 1,000.

Actually, less than that because there are two possible sets. Is it just halved, so 3 in 500?

Seems reasonable, and yes roughly twice that for both sets - so just under 7 in 1000.