What is the probability of getting 4 21's from one hand of blackjack by splitting aces followed by four tens?

Question

What is the probability of getting 4 21's from one hand of blackjack by splitting aces followed by four tens?

Answer 1

To give an accurate answer requires us to know how many shoes are in play.  But here's the general idea for a single deck, just adjust the numerators and denominators for multiple deck shoes:

Chance of first ace: 4/52

Chance of second ace: 3/51

Chance of third ace: 2/50

Chance of fourth ace: 1/49

Multiply those together to get your probability of just four aces, which comes out to 24/6,497,400 = 1/270,725.

Now, if you wanted to get the chances of now being dealt four tens (although any face card would work) pick up where you left off.

First ten = 4/48

Second = 3/47

Third = 2/46

Fourth = 1/45

To get a total of 24/4,669,920 = 1/194,580

Multiply the two together to get your final probability

1 / 52,677,670,500

One in 52 and a half billion, give or take.  And that's only with one deck, in a multi deck shoe your odds get worse.

Edit-

I was wrong about the odds in a multi deck shoe.  I ran the numbers for a six deck shoe, and your odds actually improve to 1 in 110,124,269.  Only one in a hundred and ten million!  Still pretty unlikely.

Second Edit-

I did mention that any face card would work, but the question specifically asked about tens, and I'm answering it as posted.  Once again, simply altering the numerators and denominators to account for the extra cards is all that's required in the math I've given.

Answer 2

It would never happen, as when you split aces you ar not allowed to draw aditiona cards onto them or split again.

However if you were to play somewhere that did actually allow resplitting aces, which is very rare but not unheard of.

The actualy odds would vary DRAMATICALLY based on the number of decks the cards were coming from.  since in one deck you would have to draw 100% of the aces  where in six deck you would only draw 1/6th fo the avalable aces.

Answer 3

You're probably more likely to be killed on the way to the casino.

It's going to be a pretty rare occurrence.  You'd need to split aces, get dealt two more aces, then get a ten for each ace.  You'd have to be in a casino that allows aces to be split multiple times, and even then, it wouldn't count as blackjack in most casinos, so provided the dealer didn't get 21, you'd win four times your bet.  So it would hardly be exciting financially.

I think the actual odds would depend on which set of rules the casino was using. It if were single deck blackjack you would need to get all four aces from the deck.  But if there were multiple decks there are more aces you can have.

I'm not a math wizard, but I think that certain factors like that would impact the odds of this happening.  Especially since a multi deck game may already be in progress and the shoe could be ace heavy or ace light.

EDIT: I think BigSky is a little bit wrong.  But I guess it depends if the question means specifically being dealt tens, or being dealt cards worth ten.  Let's face it, tens and faces are the same in Blackjack.  Therefore there are far more 'outs' once the four aces have come.  So it would be more like 16/48, 15/47 etc.

But in any case you are getting into lottery kind of odds, and like I said earlier the best you can win is four times your bet.  You only have to beat odds of one in 38 to get 35 times your money in Roulette.

And if you are going to try and beat odds of over 100,000,000-1, you might as well play Powerball.