Risk of ruin in blackjack (Appendix 12)

Last update: Oct. 1, 2008

There are some sources that address the question of the probability of doubling a bankroll before losing it, in a card counting situation. Ken Uston's Million Dollar Blackjack, to name one. This appendix shall not recover that issue. However ,I am often asked about how much the basic strategy player's bankroll should be, given a targeted number of hands to play. This is especialy practical if the player must play a certain number of hands to earn an online casino bonus.

The rules assumed for these tables are six decks, dealer stands on soft 17, player may double on any two cards, player may double after splitting, player may resplit to three hands, no surrender, dealer peeks for blackjack. Under these rules, the house edge is 0.4140%.

Let's look at an example of how this table can be used. Assume the player makes a deposit of $1000 to an online casino, and is required to bet through $5000 in action. If the player is to willing to play through 500 hands, then his average bet size would be $5000/500 = $10. The number of betting units would be $1000/$10 = 100. The table shows the risk of ruin is 0.01% for 102 units, so would be just over 0.01% for 100. Perhaps this is too conservative, so the player considers playing 200 hands. The bet size is now $5000/200 = $25. The number of units is $1000/$25 = 40. Interpolating the table shows the risk of ruin would be 1.5%.

Risk of Ruin	Number of Hands to Play
Risk of Ruin	100	200	300	400	500	600	700	800	900
50%	7	11	14	16	18	20	22	24	25
40%	9	14	17	20	23	25	27	29	31
30%	12	17	21	25	28	31	33	36	38
20%	15	21	26	31	34	38	41	44	47
10%	19	27	34	39	44	48	53	57	60
5%	22	32	40	46	52	58	62	67	71
4%	23	34	42	49	55	60	65	70	75
3%	25	36	44	51	58	64	69	74	79
2%	27	38	47	55	62	68	74	79	84
1%	29	42	52	61	68	75	82	88	93
0.5%	32	46	57	66	74	82	89	95	101
0.25%	35	50	61	71	80	88	96	102	109
0.1%	38	54	67	77	87	95	104	111	118
0.01%	45	64	79	91	102	112	122	131	139

Risk of Ruin	Number of Hands to Play
Risk of Ruin	1000	1200	1400	1600	1800	2000	2500	3000
50%	27	30	32	35	37	40	45	50
40%	33	37	40	43	46	49	56	62
30%	41	45	49	53	56	60	68	75
20%	50	55	60	65	69	73	83	92
10%	64	70	76	82	88	93	105	116
5%	76	83	90	97	104	110	124	137
4%	79	87	95	102	108	114	129	143
3%	83	92	100	107	114	121	136	151
2%	89	98	107	114	122	129	145	161
1%	99	108	118	126	134	142	160	177
0.5%	107	118	128	137	146	154	174	192
0.25%	115	126	137	147	156	166	187	206
0.1%	125	138	149	160	170	180	202	223
0.01%	148	162	175	188	198	212	236	261

Methodology

The tables above were created by random simulation. I have been asked several times for a general formula for other situations. Unfortunately there isn't any that I know of. Risk of ruin problems are mathematically usually very complicated. It is easier and more convincing to run a random simulation instead.

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