Killer Cards Revisited

Super/System 2 contains a chart called Lyle Berman's Killer Cards Chart (page 461). This chart explores a flaw in simple outs: just because you hit your hand doesn't mean your opponent won't make a better one.

Mr. Berman's explanation of his chart was worrying. He doesn't show his work, and wryly comments that he doesn't believe he could recreate it. (He was apparently bored enough to do it once, but not twice.) I understand not wanting to grind out that work, but why not pour some coffee down Mike Caro and see what he can do?

I decided to recreate the table. My numbers match his, which is good. I generated a complete chart for Hold 'Em as well as Omaha. I also did it a little more precisely, and then exactly: the first charts are percentages, then exact odds.

I believe this chart is correct, but it is not necessarily adequate to cover actual situations. Runner-runner events (the turn completes my flush against your set, but the river pairs the board filling you up) are not adequately explored by this table, nor are split pots. For cases where you're trying to work that out that exact math, I recommend twodimes or pokersource's poker-eval library.

Slightly edited source is available. It's rather pedestrian Common LISP (which makes the fractional math below easy). This page is the slightly edited output of this program.

Hold 'Em (45 unseen cards)

outs	0 kills	1 kills	2 kills	3 kills	4 kills	5 kills	6 kills	7 kills	8 kills	9 kills	10 kills
1	4.4444%	4.3434%	4.2424%	4.1414%	4.0404%	3.9394%	3.8384%	3.7374%	3.6364%	3.5354%	3.4343%
2	8.7879%	8.5859%	8.3838%	8.1818%	7.9798%	7.7778%	7.5758%	7.3737%	7.1717%	6.9697%	6.7677%
3	13.03%	12.727%	12.424%	12.121%	11.818%	11.515%	11.212%	10.909%	10.606%	10.303%	10.0%
4	17.172%	16.768%	16.364%	15.96%	15.556%	15.152%	14.747%	14.343%	13.939%	13.535%	13.131%
5	21.212%	20.707%	20.202%	19.697%	19.192%	18.687%	18.182%	17.677%	17.172%	16.667%	16.162%
6	25.152%	24.545%	23.939%	23.333%	22.727%	22.121%	21.515%	20.909%	20.303%	19.697%	19.091%
7	28.99%	28.283%	27.576%	26.869%	26.162%	25.455%	24.747%	24.04%	23.333%	22.626%	21.919%
8	32.727%	31.919%	31.111%	30.303%	29.495%	28.687%	27.879%	27.071%	26.263%	25.455%	24.646%
9	36.364%	35.455%	34.545%	33.636%	32.727%	31.818%	30.909%	30.0%	29.091%	28.182%	27.273%
10	39.899%	38.889%	37.879%	36.869%	35.859%	34.848%	33.838%	32.828%	31.818%	30.808%	29.798%
11	43.333%	42.222%	41.111%	40.0%	38.889%	37.778%	36.667%	35.556%	34.444%	33.333%	32.222%
12	46.667%	45.455%	44.242%	43.03%	41.818%	40.606%	39.394%	38.182%	36.97%	35.758%	34.545%
13	49.899%	48.586%	47.273%	45.96%	44.646%	43.333%	42.02%	40.707%	39.394%	38.081%	36.768%
14	53.03%	51.616%	50.202%	48.788%	47.374%	45.96%	44.545%	43.131%	41.717%	40.303%	38.889%
15	56.061%	54.545%	53.03%	51.515%	50.0%	48.485%	46.97%	45.455%	43.939%	42.424%	40.909%
16	58.99%	57.374%	55.758%	54.141%	52.525%	50.909%	49.293%	47.677%	46.061%	44.444%	42.828%
17	61.818%	60.101%	58.384%	56.667%	54.949%	53.232%	51.515%	49.798%	48.081%	46.364%	44.646%
18	64.545%	62.727%	60.909%	59.091%	57.273%	55.455%	53.636%	51.818%	50.0%	48.182%	46.364%
19	67.172%	65.253%	63.333%	61.414%	59.495%	57.576%	55.657%	53.737%	51.818%	49.899%	47.98%
20	69.697%	67.677%	65.657%	63.636%	61.616%	59.596%	57.576%	55.556%	53.535%	51.515%	49.495%
21	72.121%	70.0%	67.879%	65.758%	63.636%	61.515%	59.394%	57.273%	55.152%	53.03%	50.909%

Omaha (High, 41 unseen cards)

outs	0 kills	1 kills	2 kills	3 kills	4 kills	5 kills	6 kills	7 kills	8 kills	9 kills	10 kills
1	4.878%	4.7561%	4.6341%	4.5122%	4.3902%	4.2683%	4.1463%	4.0244%	3.9024%	3.7805%	3.6585%
2	9.6341%	9.3902%	9.1463%	8.9024%	8.6585%	8.4146%	8.1707%	7.9268%	7.6829%	7.439%	7.1951%
3	14.268%	13.902%	13.537%	13.171%	12.805%	12.439%	12.073%	11.707%	11.341%	10.976%	10.61%
4	18.78%	18.293%	17.805%	17.317%	16.829%	16.341%	15.854%	15.366%	14.878%	14.39%	13.902%
5	23.171%	22.561%	21.951%	21.341%	20.732%	20.122%	19.512%	18.902%	18.293%	17.683%	17.073%
6	27.439%	26.707%	25.976%	25.244%	24.512%	23.78%	23.049%	22.317%	21.585%	20.854%	20.122%
7	31.585%	30.732%	29.878%	29.024%	28.171%	27.317%	26.463%	25.61%	24.756%	23.902%	23.049%
8	35.61%	34.634%	33.659%	32.683%	31.707%	30.732%	29.756%	28.78%	27.805%	26.829%	25.854%
9	39.512%	38.415%	37.317%	36.22%	35.122%	34.024%	32.927%	31.829%	30.732%	29.634%	28.537%
10	43.293%	42.073%	40.854%	39.634%	38.415%	37.195%	35.976%	34.756%	33.537%	32.317%	31.098%
11	46.951%	45.61%	44.268%	42.927%	41.585%	40.244%	38.902%	37.561%	36.22%	34.878%	33.537%
12	50.488%	49.024%	47.561%	46.098%	44.634%	43.171%	41.707%	40.244%	38.78%	37.317%	35.854%
13	53.902%	52.317%	50.732%	49.146%	47.561%	45.976%	44.39%	42.805%	41.22%	39.634%	38.049%
14	57.195%	55.488%	53.78%	52.073%	50.366%	48.659%	46.951%	45.244%	43.537%	41.829%	40.122%
15	60.366%	58.537%	56.707%	54.878%	53.049%	51.22%	49.39%	47.561%	45.732%	43.902%	42.073%
16	63.415%	61.463%	59.512%	57.561%	55.61%	53.659%	51.707%	49.756%	47.805%	45.854%	43.902%
17	66.341%	64.268%	62.195%	60.122%	58.049%	55.976%	53.902%	51.829%	49.756%	47.683%	45.61%
18	69.146%	66.951%	64.756%	62.561%	60.366%	58.171%	55.976%	53.78%	51.585%	49.39%	47.195%
19	71.829%	69.512%	67.195%	64.878%	62.561%	60.244%	57.927%	55.61%	53.293%	50.976%	48.659%
20	74.39%	71.951%	69.512%	67.073%	64.634%	62.195%	59.756%	57.317%	54.878%	52.439%	50.0%
21	76.829%	74.268%	71.707%	69.146%	66.585%	64.024%	61.463%	58.902%	56.341%	53.78%	51.22%

Hold 'Em (45 unseen cards)

outs	0 kills	1 kills	2 kills	3 kills	4 kills	5 kills	6 kills	7 kills	8 kills	9 kills	10 kills
1	2/45	43/990	7/165	41/990	4/99	13/330	19/495	37/990	2/55	7/198	17/495
2	29/330	17/198	83/990	9/110	79/990	7/90	5/66	73/990	71/990	23/330	67/990
3	43/330	7/55	41/330	4/33	13/110	19/165	37/330	6/55	7/66	17/165	1/10
4	17/99	83/495	9/55	79/495	7/45	5/33	73/495	71/495	23/165	67/495	13/99
5	7/33	41/198	20/99	13/66	19/99	37/198	2/11	35/198	17/99	1/6	16/99
6	83/330	27/110	79/330	7/30	5/22	73/330	71/330	23/110	67/330	13/66	21/110
7	287/990	28/99	91/330	133/495	259/990	14/55	49/198	119/495	7/30	112/495	217/990
8	18/55	158/495	14/45	10/33	146/495	142/495	46/165	134/495	26/99	14/55	122/495
9	4/11	39/110	19/55	37/110	18/55	7/22	17/55	3/10	16/55	31/110	3/11
10	79/198	7/18	25/66	73/198	71/198	23/66	67/198	65/198	7/22	61/198	59/198
11	13/30	19/45	37/90	2/5	7/18	17/45	11/30	16/45	31/90	1/3	29/90
12	7/15	5/11	73/165	71/165	23/55	67/165	13/33	21/55	61/165	59/165	19/55
13	247/495	481/990	26/55	91/198	221/495	13/30	208/495	403/990	13/33	377/990	182/495
14	35/66	511/990	497/990	161/330	469/990	91/198	49/110	427/990	413/990	133/330	7/18
15	37/66	6/11	35/66	17/33	1/2	16/33	31/66	5/11	29/66	14/33	9/22
16	292/495	284/495	92/165	268/495	52/99	28/55	244/495	236/495	76/165	4/9	212/495
17	34/55	119/198	289/495	17/30	272/495	527/990	17/33	493/990	238/495	51/110	221/495
18	71/110	69/110	67/110	13/22	63/110	61/110	59/110	57/110	1/2	53/110	51/110
19	133/198	323/495	19/30	304/495	589/990	19/33	551/990	266/495	57/110	247/495	95/198
20	23/33	67/99	65/99	7/11	61/99	59/99	19/33	5/9	53/99	17/33	49/99
21	119/165	7/10	112/165	217/330	7/11	203/330	98/165	63/110	91/165	35/66	28/55

Omaha (High, 41 unseen cards)

outs	0 kills	1 kills	2 kills	3 kills	4 kills	5 kills	6 kills	7 kills	8 kills	9 kills	10 kills
1	2/41	39/820	19/410	37/820	9/205	7/164	17/410	33/820	8/205	31/820	3/82
2	79/820	77/820	15/164	73/820	71/820	69/820	67/820	13/164	63/820	61/820	59/820
3	117/820	57/410	111/820	27/205	21/164	51/410	99/820	24/205	93/820	9/82	87/820
4	77/410	15/82	73/410	71/410	69/410	67/410	13/82	63/410	61/410	59/410	57/410
5	19/82	37/164	9/41	35/164	17/82	33/164	8/41	31/164	15/82	29/164	7/41
6	45/164	219/820	213/820	207/820	201/820	39/164	189/820	183/820	177/820	171/820	33/164
7	259/820	63/205	49/164	119/410	231/820	56/205	217/820	21/82	203/820	49/205	189/820
8	73/205	71/205	69/205	67/205	13/41	63/205	61/205	59/205	57/205	11/41	53/205
9	81/205	63/164	153/410	297/820	72/205	279/820	27/82	261/820	63/205	243/820	117/410
10	71/164	69/164	67/164	65/164	63/164	61/164	59/164	57/164	55/164	53/164	51/164
11	77/164	187/410	363/820	88/205	341/820	33/82	319/820	77/205	297/820	143/410	55/164
12	207/410	201/410	39/82	189/410	183/410	177/410	171/410	33/82	159/410	153/410	147/410
13	221/410	429/820	104/205	403/820	39/82	377/820	91/205	351/820	169/410	65/164	78/205
14	469/820	91/164	441/820	427/820	413/820	399/820	77/164	371/820	357/820	343/820	329/820
15	99/164	24/41	93/164	45/82	87/164	21/41	81/164	39/82	75/164	18/41	69/164
16	26/41	126/205	122/205	118/205	114/205	22/41	106/205	102/205	98/205	94/205	18/41
17	136/205	527/820	51/82	493/820	119/205	459/820	221/410	85/164	102/205	391/820	187/410
18	567/820	549/820	531/820	513/820	99/164	477/820	459/820	441/820	423/820	81/164	387/820
19	589/820	57/82	551/820	133/205	513/820	247/410	95/164	114/205	437/820	209/410	399/820
20	61/82	59/82	57/82	55/82	53/82	51/82	49/82	47/82	45/82	43/82	1/2
21	63/82	609/820	147/205	567/820	273/410	105/164	126/205	483/820	231/410	441/820	21/41

Tim Showalter / tjs@psaux.com