“The Hand” - Split and Double

[Mr. Ed]

When I was first beginning my AP career, the first book I ever read referred to “The Illustrious 18”, authored by some clever, perceptive numbers guy - able to distill a huge amount of information into a usable, simple, “rule of thumb”.

Then I found out he authored a book, which many in the AP community referred to as “The Bible”, I had to have it. It contained a wealth of information, informative, detailed, accurate. I was in mathematical heaven.

What struck me first and foremost was Chapter 1 - “The Hand”. The author had a max bet out on two hands. One of them split and doubled. He ended up losing all of those bets, but came back (eventually) to be far ahead.

That was the day that I decided, “When I play for real, I will always keep two max bets in reserve, so when I face “The Hand” I can go down with dignity and confidence, ready to eventually come back a winner.”

I hope this is a good, useful, “rule of thumb” for all of you APs.

P.s. When I dove into the numbers a little deeper, I realized that most splits (more than 50%) will require an additional bet, either double or split. For example, for TC>=2, 33v6 (the example in the book) will require an additional bet 72% of the time!!!

[Mr. Ed]

P.p.s. To be precise…Spitting 22 through 77 requires an additional bet more than 50% of the time, 88 and 99 less than 50%, AA no further splits (usually) permitted. And just don’t split 10s. (Except 77v7, <50%)

[aceside]

If we split about 2% of all hands and double about 10% of them, we split+double about 2%x10%=0.2% of all hands.
However, when we split, we would be more likely to double on these split hands, so the split+double hands should be >>0.2%. We need Don to estimate this.

[aceside]

If we split about 2% of all hands and double about 10% of them, we split+double about 2%x10%=0.2% of all hands.
However, when we split, we would be more likely to double on these split hands, so the split+double hands should be >>0.2%. We need Don to estimate this.

I just noticed a no helpful comment and thus revise my math now.

If we split about 2% of all rounds and double about 10% of each of the split hands, we split+double about 2x2%x10%=0.4% of all rounds, that is, one in every 250 rounds.
However, when we split, we would be more likely to double on these split hands, so the split+double rounds should be >>0.4%.

[Mr. Ed]

Ok. Here is my best estimate of the probabilities of needing at least 2 additional bets:
1 hand: 0.9% (1 in 109)
2 hands: 3.31% (1 in 30)

So Aceside is correct - 0.9% is greater than 0.4%.

Assumptions: TC=0, DAS, h17, no resplit Aces
Chance of Double: 10.1%
Chance of split: 2.59%
Chance of Double after Split: 35.4%

For 2 hands:
Chance of both doubling: 1.02%
Chance of one hand doubling and one hand split: 0.52%
Chance of at least one hand split, then double: 1.8%

[aceside]

Ok. Here is my best estimate of the probabilities of needing at least 2 additional bets:
1 hand: 0.9% (1 in 109)
2 hands: 3.31% (1 in 30)

So Aceside is correct - 0.9% is greater than 0.4%.

Assumptions: TC=0, DAS, h17, no resplit Aces
Chance of Double: 10.1%
Chance of split: 2.59%
Chance of Double after Split: 35.4%

For 2 hands:
Chance of both doubling: 1.02%
Chance of one hand doubling and one hand split: 0.52%
Chance of at least one hand split, then double: 1.8%

First of all, let us simply the question. We consider only these blackjack rounds that include one split and an additional double-down to any one of these two split hands. Let us use your above numbers to calculate the probability of such an around:

2x2.59%x35.4%=1.8%, that is one in 55 rounds.

why do you say it is 0.9%?

[Mr. Ed]

Ok. Good catch - Sorry, so I was lazy in my post. The 35.4% is not the chance of “double after split”, but the “chance that at least one of the two hands will require at least one more bet”, which includes a re-split, and subtracts the prob of both hands needing another bet.

So, 22v6 will double if you get a 7, 8, 9, or A and resplit if you get at 2, so this number is (1-(13-5)/13^2)=62%. For AAv6, it is 1-(13-0)/13^2=0%. An average of every split is 35.4%.

So I just multiplied 2.59%x35.4%=0.9%.

P.s. If you want to exclude the resplit reality, be my guest. The probability is 26.2% instead of 35.4%. I am not going to recalc those numbers because I don’t care.

P.p.s. Actually, after I posted this, I realized the actual question I was trying to answer was, “What is 1- prob of (needing THREE or more additional bets).?” But I realized this is a complicated question, and I don’t really need a super-accurate answer to play successful BJ.

[aceside]

This is reasonable. The probability of a blackjack round that include one split and an additional double-down to any one of these two split hands is 2.59%x26.2%=0.68%, that is, one in 147. We should see it once every one hour of playing.