## Sticky Thirteens

Sticky Thirteens is a card game which is played at two of my local pubs.

You are given 13 random playing cards. The dealer then takes his own complete, shuffled deck of 52 and turns them over one at a time, calling out the values. When all of your 13 cards have appeared, you win.

Obviously there will be many other players, each with their own random 13 taken from different decks (each 13-pack is marked and distributed by the pub so nobody can surreptitiously substitute their own fixed deck). So, like Bingo, it is a game of chance, you just wait and see who wins. There is no skill element.

Ordinarily the dealer just keeps going until somebody calls. In this case you are assured of an eventual winner, though there may be a tie. However, at one of the pubs I realised that a different game was being played. The dealer turned over all but 25 cards, and if nobody won, the game was over. The jackpot prize was held until the following week, when the dealer would turn over all but 24 cards, and so on, until, naturally, the number got low enough that somebody won.

I studied the game for a while, wondering what the expected winnings were if I were to enter. I tried to figure out some probabilities but didn't have the space to calculate, but alarm bells were ringing.

Eventually I got home and, the lazy bum that I am, punched the figures into an Excel spreadsheet and hit "Fill" to figure out the actual probabilities. Guess what? Nobody is going to win for a long time.

### Formula

Let's say that you start with 13 cards in your hand, and the deck has 52 cards in it.

As the game progresses, the deck will be turned over one card at a time. Let's call the number of cards remaining in the deck, A. A is 52 to begin with.

Every time a card is turned over, you may find yourself able to discard a card from your hand. Let's call the number of cards remaining in your hand B. B is 13 to begin with.

Initial conditions are (A,B) = (52,13). You are trying to get to (A,0) for some value of A.

Let's call P(A,B) the probability of arriving at situation (A,B). Since we know we start at (52,13) we know that P(52,13)=1.

Let's say for the sake of argument that the dealer keeps turning cards until all 52 have been revealed, regardless of what happens. By the end of the deck, you know that all the cards in your hand will have been discarded. So, we know that P(0,0)=1 too.

I won't bother you with the details of the tedious calculation which led to this formula, but I found that in general:

```                39! 13! (52-A)! A!
P(A,B) = ---------------------------------
52! (A-B)! B! (13-B)! (39-A+B)!
```

### Table

This gives the following table of probabilities:

P(A,B)Number of cards remaining in hand (B)
131211109876543210
Number of cards remaining in deck (A)521.00000
510.750000.25000
500.558820.382350.05882
490.413530.435880.137650.01294
480.303820.438850.213490.041200.00264
470.221530.411420.274280.081540.010730.00050
460.160260.367650.315130.128390.026020.002470.00008
450.114970.317030.335680.175830.048840.007130.000500.00001
440.081750.265700.338170.218810.078150.015630.001690.000090.00000
430.057600.217390.326090.253630.111890.028770.004260.000350.000010.00000
420.040190.174140.303340.278060.147460.046840.008920.000990.000060.000000.00000
410.027750.136820.273650.291300.182060.069520.016360.002340.000190.000010.000000.00000
400.018950.105580.240280.293670.213150.095920.027130.004790.000510.000030.000000.000000.00000
390.012790.080060.205870.286330.238610.124690.041560.008820.001170.000090.000000.000000.000000.00000
380.008530.059690.172440.270980.256960.154180.059680.014920.002370.000230.000010.000000.000000.00000
370.005610.043760.141370.249590.267410.182580.081150.023560.004420.000520.000040.000000.000000.00000
360.003640.031540.113540.224160.269820.208150.105270.035090.007640.001060.000090.000000.000000.00000
350.002320.022340.089360.196580.264630.229350.131060.049710.012430.002000.000200.000010.000000.00000
340.001460.015540.068930.168500.252750.244970.157270.067400.019170.003550.000410.000030.000000.00000
330.000900.010620.052120.141240.235410.254240.182530.087880.028250.005950.000790.000060.000000.00000
320.000550.007110.038600.115810.214010.256810.205450.110620.039950.009510.001440.000130.000010.00000
310.000320.004670.028000.092890.190010.252790.224710.134820.054450.014560.002500.000260.000010.00000
300.000190.003000.019870.072860.164810.242720.239200.159470.071760.021470.004130.000480.000030.00000
290.000110.001880.013780.055860.139660.227440.248110.183390.091690.030560.006580.000860.000060.00000
280.000060.001150.009330.041830.115580.208040.250970.205340.113830.042160.010120.001490.000120.00000
270.000030.000680.006160.030560.093370.185750.247660.224080.137500.056460.015060.002460.000220.00001
260.000020.000400.003950.021750.073560.161830.238490.238490.161830.073560.021750.003950.000400.00002
250.000010.000220.002460.015060.056460.137500.224080.247660.185750.093370.030560.006160.000680.00003
240.000000.000120.001490.010120.042160.113830.205340.250970.208040.115580.041830.009330.001150.00006
230.000000.000060.000860.006580.030560.091690.183390.248110.227440.139660.055860.013780.001880.00011
220.000000.000030.000480.004130.021470.071760.159470.239200.242720.164810.072860.019870.003000.00019
210.000000.000010.000260.002500.014560.054450.134820.224710.252790.190010.092890.028000.004670.00032
200.000000.000010.000130.001440.009510.039950.110620.205450.256810.214010.115810.038600.007110.00055
190.000000.000000.000060.000790.005950.028250.087880.182530.254240.235410.141240.052120.010620.00090
180.000000.000000.000030.000410.003550.019170.067400.157270.244970.252750.168500.068930.015540.00146
170.000000.000000.000010.000200.002000.012430.049710.131060.229350.264630.196580.089360.022340.00232
160.000000.000000.000000.000090.001060.007640.035090.105270.208150.269820.224160.113540.031540.00364
150.000000.000000.000000.000040.000520.004420.023560.081150.182580.267410.249590.141370.043760.00561
140.000000.000000.000000.000010.000230.002370.014920.059680.154180.256960.270980.172440.059690.00853
130.000000.000000.000000.000000.000090.001170.008820.041560.124690.238610.286330.205870.080060.01279
120.000000.000000.000000.000030.000510.004790.027130.095920.213150.293670.240280.105580.01895
110.000000.000000.000010.000190.002340.016360.069520.182060.291300.273650.136820.02775
100.000000.000000.000060.000990.008920.046840.147460.278060.303340.174140.04019
90.000000.000010.000350.004260.028770.111890.253630.326090.217390.05760
80.000000.000090.001690.015630.078150.218810.338170.265700.08175
70.000010.000500.007130.048840.175830.335680.317030.11497
60.000080.002470.026020.128390.315130.367650.16026
50.000500.010730.081540.274280.411420.22153
40.002640.041200.213490.438850.30382
30.012940.137650.435880.41353
20.058820.382350.55882
10.250000.75000
01.00000

These figures are to five decimal places. Several probabilities are so low that they register as 0 on this scale, however, only the blank cells represent actually impossible events.

### Simplified formula

The important column is, as I've said, P(A,0), which is the far right side.

Setting B=0 allows us to simplify the formula to:

```          39! (52-A)!
P(A,0) = -------------
52! (39-A)!
```

### Simplified table

The right two columns of this assume 15 players each week.

Undrawn cards (A)P(A,0)P(no one wins)P(no one has won yet)
52011
51011
50011
49011
48011
47011
46011
45011
44011
43011
42011
41011
40011
390.000001.000001.00000
380.000001.000001.00000
370.000001.000001.00000
360.000001.000001.00000
350.000001.000001.00000
340.000001.000001.00000
330.000001.000001.00000
320.000001.000001.00000
310.000001.000000.99999
300.000000.999990.99998
290.000000.999970.99995
280.000000.999940.99989
270.000010.999880.99977
260.000020.999750.99953
250.000030.999530.99905
240.000060.999120.99817
230.000110.998400.99657
220.000190.997170.99376
210.000320.995140.98892
200.000550.991830.98084
190.000900.986550.96764
180.001460.978300.94665
170.002320.965690.91417
160.003640.946780.86552
150.005610.919070.79548
140.008530.879450.69958
130.012790.824400.57674
120.018950.750530.43286
110.027750.655670.28381
100.040190.540510.15340
90.057600.410700.06300
80.081750.278210.01753
70.114970.160100.00281
60.160260.072810.00020
50.221530.023370.00000
40.303820.004370.00000
30.413530.000330.00000
20.558820.000000.00000
10.750000.000000.00000
01.000000.000000.00000

In other words, at A=25, the odds of a given person winning are around 32,000 to 1! It looks like we're going to be here until we get down to around 14 cards at the earliest, or another 10 weeks. Still, by that time, maybe the jackpot will be large enough to make the investment worthwhile, hmm? I wonder if there's a limit on how many times you can enter each week.

### Generalised formula

For a deck of size X and a hand of size Y, the formula is:

```                 (X-Y)! Y! (X-A)! A!
P(A,B) = ------------------------------------
X! (A-B)! B! (Y-B)! ((X-Y)-(A-B))!
```