Blackjack odds of hitting 21 or busting...

[oddsock]

Hi,

I am wondering if anyone can give me pointers to a site where I can get odds of making 21 with all 2-card starting hands or even help me on my way of working this out. I've checked the Wizard of Odds site but to no avail.

It's a single deck game and no other cards are visible.

Thanks.

[Gullanian]

1 pack of cards

chance of hitting a '10' is 16/52
chance of hitting an ace is 4/52

being dealt a BJ = 16/52 * 1/13

[AaronBrown]

It's a bit tedious to work out, but not difficult in princple. I assume you keep hitting until you get 21 or bust, obviously it would change the probability if you played differently, say by stopping when you got 17 or higher.

Let's start with an infinite deck, which makes the problem much easier. Let P(c) be the probability of hitting 21 before busting if your current count is c.

P(21) = 1
P(20) = P(21)/13 = 1/13
P(19) = P(20)/13 + P(21)/13 = 14/169

and so on for hard counts. For a soft count you have:

PS(21) = 1
PS(20) = PS(21)/13 + 4*P(20)/13 = 1/13 + 4/169 = 17/169
PS(19) = PS(21)/13 + PS(20)/13 + 4*P(19)/13

and so on.

I used a spreadsheet to get the probabilities of hitting 21 with hard count (middle column) and soft count (right column). I don't guarantee it's correct.

21 100.00% 100.00%
20 7.69% 10.06%
19 8.28% 11.02%
18 8.92% 12.06%
17 9.61% 13.20%
16 10.35% 14.44%
15 11.14% 15.80%
14 12.00% 17.27%
13 12.92% 18.89%
12 13.92% 20.65%
11 38.06% 41.03%
10 12.82% 16.51%
9 14.04% 18.07%
8 15.34% 19.79%
7 16.77% 21.66%
6 18.32% 23.71%
5 20.01% 25.95%
4 21.86% 28.40%
3 #N/A 31.08%
2 #N/A 34.01%

With a single deck, you'll need to use a computer program to add up all the possibilities. The trouble is that each card changes the probabilities of subsequent cards. The easiest way is to consider all possible deals from AK to AAAA2222333.