Bonus for Suited Blackjack

I would like to see if someone can verify my calculations for the percent gain in expectation for receiving a 2 for 1 payoff on suited blackjacks in a single deck game.

The probability of receiving a blackjack is 128/2652 or 4.8265%
Thus, we will receive a suited blackjack 1/4 of the time or 1.2066%
Because the dealer will also receive blackjacks some of the time, we need to subtract from this advantage for the times the dealer receives a BJ.
A dealer will receive a blackjack the same round we do 128/2652 times 90/2460 of the time or .1766%

This reduces our probability to 1.0301% and with a .5 units bonus our expectation is increased by .5150%

Please let me know if I have made any errors in my calculations.

As this is my first post, I hope I have not made any errors.

John

Greetings, John.

It's always a little tricky when you try to add or subtract probabilities. You have to make sure you're comparing apples to apples, and all the fruit are mutually exclusive. ;-) Your figure doesn't quite gel with other published figures, and I think the discrepancy comes from the part where you subtract the prob both dealer and player have a BJ from the prob of a suited BJ. If you rethink the assumptions underlying that step, I think you'll be gratified by the result.

L:U:C:K
ETF

Thanks for the help.

I think I understand. The probability of the dealer receiving a blackjack and the probability of the player receiving a suited BJ is 128/2652 times 1/4 times 90/2460 which is .0441%.

Correct?

I am not sure if the probability of a dealer blackjack, given a player has a BJ is correct 90/2460?

John

It is well known that a 2-1 BJ payoff is an additional 2.3% player edge over the usual 3-2 payoff. It will be suited 1/4 of the time, so you would expect a little under 0.6% gain.

As you seem to take the game & math seriously, you should consider visiting Richard Reid's site, www.bjmath.com , or join green chip and post such discussions on the theory page. I doubt that you will get very detailed responses on misc free pages. Most discussions here are very general.

a * t * (1-2(a-1)*(t-1)/(n-2)/(n-3))/n/(n-1)
where a=number of aces, t=number of tens, and n=cards remaining.
from page 191 of Peter Griffin's "The Theory of Blackjack".

Plug in a=4,t=16,and n=52 and you get the probability of winning with a blackjack. Divide by 4 and that is winning with a suited blackjack.

John wrote:

"I am not sure if the probability of a dealer blackjack, given a player has a BJ is correct 90/2460?"

My guess is that you understand how to compute the probability, but that you have a typo here. The denominator is 50x49, or 2450. I can't reconstruct any incorrect path of reasoning that would lead you to 2460.

Here's my take on the problem:
Chance of player blackjack = 128/2652 = 0.048265
Chance of dealer blackjack given that player has blackjack = 90/2450 = 0.036735
Chance of dealer nonblackjack given that player has blackjack = 1 - 0.036735 = 0.963265 (i.e. more than 96% of your blackjacks will be winners)
Chance of player having winning blackjack = 0.048265 x 0.963265 = 0.046492
Chance of player having winning suited blackjack = 0.046492/4 = 0.011623
Gain to player from winning an extra half unit on winning suited blackjacks = 0.011623/2 = 0.005812 = .5812%

Seeker

Thanks for the help.

My pleasure.

I think I understand. The probability of the dealer receiving a blackjack and the probability of the player receiving a suited BJ is 128/2652 times 1/4 times 90/2460 which is .0441%.

Correct?

Awwwwwwful close! I wouldn't mark it wrong in an exam.

I am not sure if the probability of a dealer blackjack, given a player has a BJ is correct 90/2460?

Check the denominator again, and don't peek at Seeker's answer! Takes away all the fun just to hand out answers, don't you agree? Tsk, tsk, seeker. ;~)

Here's another question for you. You say that the probability of a suited BJ is 1/4 the prob of any old BJ. Now, what's the prob of drawing ANY two suited cards at random. Is it 1/4? If not, why not?

ETF

I knew the denominator of the prob of dealer BJ given player BJ was 50*49. I guess I read it wrong off of my calculator. A six looks like a 5. 2450 vs 2460.

With that slight correction, my answer comes out exactly correct, given the formula in Griffin's book. I have a copy of griffin's book but I didn't remember seeing a calculation for the gain from BJ in the book.

To answer your last question, I have worked through this suited problem before (Royal Match). It is less than 1/4. However intuition leaves us to believe it is 1/4 and that is why so many people bet on Royal Match incorrectly thinking it is an even bet.

Thanks for all your help.

John

Now that I have the "correct" answer, I can be safe in saying that this game has a positive expectation w/out counting.

Single Deck, DOA, H17, BJ2:1 Suited Pair = -.19%+.58% = +.39%.

Unfortunately this game is only offered on Friday Nights from 6pm to Midnight where I play.

John

Well, John, now we know where you play. I will email you for additional discussion. Bonne chance. PV