A game called Tonk

[kilfin]

There is a game that people play where I live known as Tonk. I'm unsure as to whether or not it is widely played or not...its very much like rummy...but with some twist. If anyone is interested in the rules I'd be happy to give them to you, but I basically have just one question that has to do with figuring out some odds for the game. My question is this:

The value of the cards are below:

A = 1
10, J, Q, K = 10
The rest are equal to face value (e.g. 2 = 2, 3 = 3, etc)

I would like to know what the lowest hand dealt will be 70% of the time with three people playing, four people playing, and five people playing. I know I'm not doing a very good job of putting it into words but its something like this.

If there are three people playing and I am dealt 2-5-A-A-3 that adds up to 12. From my experience that is a winner nearly 100% of the time, of course that doesn't happen very often. It might be better to put it this way....25 points will win x% of the time with x-number of people playing considering you are first and nobody has had a chance to draw and reduce points. How would I go about figuring this out.

One more thing...the reason I need to figure out what will win 70% of the time is because if I lay down with three people playing and I win, I collect one big bet from every player, If I lay down and I am beat or someone ties me, I have to pay out one big to the winner bet for every player at the table, including myself. So if the bets are $1 and $2 and I win by laying down in the first round I would win $4...$2 from each of the other players (If three people are playing), but if I lose I would have to pay out $6 to the winner. If i were to stick to the 70% rule I'm trying to figure out I would come out ahead in the long run. If someone could tell me how to figuree this out...it would be much appreciated. If you need more explanation or have a question just ask...I think I explained everything thats needed to figure it out but I could be wrong.

I would like to figure out how to make a chart something like this

% of chance to win

Points 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16

P 3
l 4
a 5
y
e
r
s

[Phat Mack]

[ QUOTE ]
One more thing...the reason I need to figure out what will win 70% of the time is because if I lay down with three people playing and I win, I collect one big bet from every player, If I lay down and I am beat or someone ties me, I have to pay out one big to the winner bet for every player at the table, including myself. So if the bets are $1 and $2 and I win by laying down in the first round I would win $4...$2 from each of the other players (If three people are playing), but if I lose I would have to pay out $6 to the winner.

[/ QUOTE ]

If you are against 2 opponents, you are laying 6:4, so in order to break even you need to win 6/10 = 60% of the time. You will win $4*6=$24, and lose $6*4=$24.

If you are against 3 opponents, you are laying 8:6. In order to break even, you need to win 8/14=57%. Win $6*8=$48; lose $8*6=$48.

If you are against 4 opponents, you are laying 10:8. In order to break even, yon need to win 10/18= 56% of the time.

---------------

OK. If you were against 1 opponent and wanted to win 60% of the time, you could make a list of every single tonk hand, rank them in point-count order, and only dive with those hands better than the 60th percentile. Against two opponents, they both have to lose 60% of the time, so lets take .60^1/2 = .775 percentile hands. That gives us a 27 from our list, so we'll play a 26.

Against three opponents we need a .57^1/3= .829 percentile hand, which is a 26. Therefore we can play 25's.

Against four opponents we need to beat a .56^1/4= .863 percentile hand, which is a 25, so we can play 24's.

To summarize, if you want to dive on the first round, and your opponents haven't played , 3-handed dive with 26; 4-handed dive with 25, 5-handed dive with 24. (What to do when your opponents draw before you act is an interesting question, but isn't covered here).

To answer your original question, if you want to up your edge to .70, play 25, 23, 22 respectively.

I've never played tonk with special rules for first-round dives. What happens if more than one person beats or ties you?

[AaronBrown]

I can't give you the complete answer, but here is some help. The table below shows the possible totals from 6 to 50, the number of ways each total can form out of the 2,598,968 possible five-card hands, and the percentage chance of getting a hand of that total or higher.

6 4 100.00%
7 28 100.00%
8 92 100.00%
9 240 100.00%
10 485 99.99%
11 920 99.97%
12 1,552 99.93%
13 2,492 99.87%
14 3,784 99.78%
15 5,725 99.63%
16 8,344 99.41%
17 11,988 99.09%
18 16,520 98.63%
19 22,144 97.99%
20 28,949 97.14%
21 36,708 96.03%
22 45,584 94.61%
23 55,712 92.86%
24 67,600 90.72%
25 79,417 88.12%
26 92,416 85.06%
27 103,808 81.50%
28 115,520 77.51%
29 125,188 73.07%
30 134,053 68.25%
31 140,224 63.09%
32 146,936 57.69%
33 149,268 52.04%
34 147,784 46.30%
35 143,677 40.61%
36 136,344 35.08%
37 127,484 29.84%
38 116,832 24.93%
39 105,176 20.44%
40 92,549 16.39%
41 82,176 12.83%
42 65,532 9.67%
43 52,556 7.15%
44 40,436 5.12%
45 31,217 3.57%
46 22,496 2.37%
47 16,720 1.50%
48 10,640 0.86%
49 7,280 0.45%
50 4,368 0.17%

You might think that playing against one player, your chance of winning or tying with a 24 is 90.72%. But it's higher than that. You have a lower than average total, meaning the cards remaining in the deck are higher than average.

This table shows the possible totals, and the chance that someone has you beaten (not just tied) if there are 2, 3, 4 or 5 players in the game (2 is the second column from the left, 5 is the column on the right). It's based on a Normal approximation and ignores dependence among the other players' hands (but does account for your cards):

6 0.00% 0.00% 0.00% 0.00%
7 0.00% 0.00% 0.00% 0.00%
8 0.00% 0.00% 0.01% 0.01%
9 0.00% 0.01% 0.01% 0.02%
10 0.01% 0.02% 0.02% 0.03%
11 0.01% 0.03% 0.04% 0.06%
12 0.03% 0.05% 0.08% 0.11%
13 0.05% 0.10% 0.15% 0.20%
14 0.09% 0.17% 0.26% 0.35%
15 0.15% 0.30% 0.45% 0.60%
16 0.25% 0.51% 0.76% 1.01%
17 0.42% 0.83% 1.24% 1.65%
18 0.66% 1.33% 1.98% 2.63%
19 1.04% 2.07% 3.08% 4.09%
20 1.58% 3.14% 4.67% 6.18%
21 2.36% 4.66% 6.90% 9.10%
22 3.42% 6.73% 9.93% 13.01%
23 4.86% 9.49% 13.89% 18.08%
24 6.75% 13.04% 18.91% 24.38%
25 9.15% 17.47% 25.02% 31.89%
26 12.14% 22.81% 32.18% 40.42%
27 15.76% 29.03% 40.22% 49.64%
28 20.02% 36.03% 48.84% 59.08%
29 24.91% 43.61% 57.65% 68.20%
30 30.36% 51.51% 66.23% 76.48%
31 36.30% 59.42% 74.15% 83.53%
32 42.58% 67.03% 81.07% 89.13%
33 49.06% 74.05% 86.78% 93.27%
34 55.56% 80.25% 91.23% 96.10%
35 61.92% 85.50% 94.48% 97.90%
36 67.97% 89.74% 96.71% 98.95%
37 73.58% 93.02% 98.15% 99.51%
38 78.63% 95.43% 99.02% 99.79%
39 83.08% 97.14% 99.52% 99.92%
40 86.88% 98.28% 99.77% 99.97%
41 90.05% 99.01% 99.90% 99.99%
42 92.61% 99.45% 99.96% 100.00%
43 94.64% 99.71% 99.98% 100.00%
44 96.20% 99.86% 99.99% 100.00%
45 97.37% 99.93% 100.00% 100.00%
46 98.22% 99.97% 100.00% 100.00%
47 98.82% 99.99% 100.00% 100.00%
48 99.24% 99.99% 100.00% 100.00%
49 99.52% 100.00% 100.00% 100.00%
50 99.71% 100.00% 100.00% 100.00%

[Phat Mack]

Very nice.