Mixing hand probabilities

Hey everyone,

The probability of getting a BJ when delt an Ace in a 6 deck shoe is 96/311 = 0.3087 or 30.87%
The probability of getting a BJ when dealt a face card in a 6 deck shoe is 24/311 = 0.077 or 7.7%

Through combinatorial analysis I know that the overall probability of getting a BJ in a 6 deck shoe is 0.04748 or 4.75%

How do I mix (add, multiply, average, combine, whatever it is) the first two numbers above to get the total of 4.75% that I know is correct from combinatorial analysis?

Thanks

I think you need to introduce the probability of being dealt an ace first (24/312) and the probability of an X first (96/312).

Take it with a grain of salt; I'm not a math guy. ;)

Quote: gunbj

Hey everyone,

The probability of getting a BJ when delt an Ace in a 6 deck shoe is 96/311 = 0.3087 or 30.87%
The probability of getting a BJ when dealt a face card in a 6 deck shoe is 24/311 = 0.077 or 7.7%

Through combinatorial analysis I know that the overall probability of getting a BJ in a 6 deck shoe is 0.04748 or 4.75%

How do I mix (add, multiply, average, combine, whatever it is) the first two numbers above to get the total of 4.75% that I know is correct from combinatorial analysis?

Thanks
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gunbj,

Two ways to get a BJ are
1. A followed by X
2. X followed by A

Since these two ways are mutually exclusive, we know:

P(BJ) = P(A then X) + P(X then A)

From a full a 6-deck game shoe:

P(A then X) = (24/312)*(96/311) = 0.023744744001978728666831...

P(X then A) = (96/312)*(24/311) = 0.023744744001978728666831...

Summing gives P(BJ) = 0.047489488003957457333663..., or about 4.75%

Note P(A then X) = P(X then A): if you look at the calculations, all we did was switch the positions for the 24 and 96, and since these are multiplied and multiplication is commutative, switching them does not change the result. Thus, the shorter way to find P(BJ) is this:

P(BJ) = 2*(24/312)*(96/311)

Hope this helps!

Dog Hand

Homework: calculate P(BJ) for a single deck game. Due tomorrow.

Quote: DogHand

Quote: gunbj

Hey everyone,

The probability of getting a BJ when delt an Ace in a 6 deck shoe is 96/311 = 0.3087 or 30.87%
The probability of getting a BJ when dealt a face card in a 6 deck shoe is 24/311 = 0.077 or 7.7%

Through combinatorial analysis I know that the overall probability of getting a BJ in a 6 deck shoe is 0.04748 or 4.75%

How do I mix (add, multiply, average, combine, whatever it is) the first two numbers above to get the total of 4.75% that I know is correct from combinatorial analysis?

Thanks
link to original post

gunbj,

Two ways to get a BJ are
1. A followed by X
2. X followed by A

Since these two ways are mutually exclusive, we know:

P(BJ) = P(A then X) + P(X then A)

From a full a 6-deck game shoe:

P(A then X) = (24/312)*(96/311) = 0.023744744001978728666831...

P(X then A) = (96/312)*(24/311) = 0.023744744001978728666831...

Summing gives P(BJ) = 0.047489488003957457333663..., or about 4.75%

Note P(A then X) = P(X then A): if you look at the calculations, all we did was switch the positions for the 24 and 96, and since these are multiplied and multiplication is commutative, switching them does not change the result. Thus, the shorter way to find P(BJ) is this:

P(BJ) = 2*(24/312)*(96/311)

Hope this helps!

Dog Hand

Homework: calculate P(BJ) for a single deck game. Due tomorrow.
link to original post

Thanks for the reply Dog Hand and for the homework ;)

What about adding the probabilities of the dealer busting with each initial card to get the total probability of the dealer busting in general?
Like in this chart:
https://www.beatblackjack.org/en/strategy/dealer-probabilities/
How do you get 28.2% from the individual numbers?

I did the homework a couple of weeks ago. Let me post what I posted.

Let me quote Eliot’s post about insurance. “For example, in a single deck game, there are 16 face cards and 35 non-face cards (the Ace is already exposed). It follows that the true odds for Insurance are 35-to-16, while it pays 32-to-16. The edge on the Insurance bet is then 3/51, or 5.8824%.”

Your above question is a little tricky, because it depends on the rule of if the dealer has a hole card or not. If not, you just do the same calculation as what the above two mathematicians told you.

Quote: gunbj

<snip>Thanks for the reply Dog Hand and for the homework ;)

What about adding the probabilities of the dealer busting with each initial card to get the total probability of the dealer busting in general?
Like in this chart:
https://www.beatblackjack.org/en/strategy/dealer-probabilities/
How do you get 28.2% from the individual numbers?
link to original post

gunbj,

Two ways to find P(bust) from the given data:

1. Since the dealer's result will be one of 17, 18, 19, 20, non-BJ 21, BJ, or bust (assuming the dealer must finish her hand, which is what that website assumes), the sum of their probs must be 100%. Add the values for the non-bust results and subtract the total from 100% to get 28.2%.

2. For each upcard, multiply the P(bust)*P(DFH), then sum these values to get the P(bust). Note that this doesn't work exactly with the numbers shown because we need more decimal places to get a more-accurate value.

By the way, that website confusingly refers to the first "hand" of the dealer, rather than the first card: hence the abbrev. DFH means dealer's first hand... Odd.

Hope this helps!

Dog Hand

P.S. Where's your homework? Don't tell me the dog ate it... That's simply anti-canine propaganda! ;-)

It would be more appropriate to link to the wizardofodds page than to an external page, especially since Mike has gone to the effort to put so much great numerical information on his various BJ pages.

https://wizardofodds.com/games/blackjack/dealer-odds-blackjack-us-rules/

Quote: DogHand

gunbj,

Two ways to find P(bust) from the given data:

1. Since the dealer's result will be one of 17, 18, 19, 20, non-BJ 21, BJ, or bust (assuming the dealer must finish her hand, which is what that website assumes), the sum of their probs must be 100%. Add the values for the non-bust results and subtract the total from 100% to get 28.2%.

Thanks, I honestly don’t know how I missed that 🤦‍♂️

Quote: DogHand

P.S. Where's your homework? Don't tell me the dog ate it... That's simply anti-canine propaganda! ;-)
link to original post

Lol. Where I’m staying right now, there’s mongooses wandering the garden, but I won’t pretend they ate it, that’be even more far fetched!

Here it is: 2*(16/52)*(4/51)= 0.048265 or 4.83%! (I hope)

Quote: Mental

It would be more appropriate to link to the wizardofodds page than to an external page, especially since Mike has gone to the effort to put so much great numerical information on his various BJ pages.

https://wizardofodds.com/games/blackjack/dealer-odds-blackjack-us-rules/

DEALER'S                              DEALER'S FINAL TOTAL
UP CARD	    17	        18	        19	        20	        21	        BUST
Ace	0.183786	0.190890	0.188680	0.191692	0.075137	0.169815
2	0.138976	0.131762	0.131815	0.123948	0.120526	0.352973
3	0.130313	0.130946	0.123761	0.123345	0.116047	0.375588
4	0.130973	0.114163	0.120679	0.116286	0.115096	0.402803
5	0.119687	0.123483	0.116909	0.104694	0.106321	0.428905
6	0.166948	0.106454	0.107192	0.100705	0.097879	0.420823
7	0.372345	0.138583	0.077334	0.078897	0.072987	0.259854
8	0.130857	0.362989	0.129445	0.068290	0.069791	0.238627
9	0.121886	0.103921	0.357391	0.122250	0.061109	0.233442
10	0.124156	0.122486	0.124421	0.356869	0.039570	0.232499
All	0.153225	0.145065	0.141657	0.184722	0.077364	0.297967

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Oops, sorry. I posted the first thing that popped up out of convenience.