What is the value(per hand) of a $500 bonus if you receive 3 blackjacks in a row, and if each consecutive blackjack after that, another $1000? Another words, if you receive 4 blackjacks in a row, you will receive 1500 total in bonuses, besides you regular 3 to 2 payoff for each blackjack(500 for the third and 100 for the fourth). Assume that a 6 deck shoe is used. What is the exact frequency of occurrence for 3 in a row;4 in a row;5 in a row?
3 in a row: ((24X96)X(23X95)X(22X94))/((312X311)X(310X309)X(308X307))=.000011845=P3
4 in a row: P3X21X93/(306X305)=.000000247=P4
5 in a row: P4X20X92/(304X303)=.000000004=P5
Each denominator above must be divided by 2^n, i.e. P3=.000094765, P4=.000003966 and P5=.0000001580.
The probability of any player natural (the poster doesn't say if we get the bonus for a tied natural or not) in a 6-deck game is 0.0475.
So, three in a row is (0.0475)^3 = 0.0001071. In other words, we will see three naturals in a row approximately once every 9,337 hands.
The value per hand to you is, of course, then simply $500/9337 = about 5 cents per hand.
Don't hold your breath!
As for the fourth natural, once you have three, the conditional probability is great: $1,000 for a 1 in 21 shot, or almost $50 EV for that next hand. The trick is to get there! :-)
Don
Once three naturals have occurred, that means at least three fewer tens and three fewer aces to make another natural. Of course, in a six-decker, this might not mean much, but if the distribution of the cards has been fairly even prior to the three BJ's, I wonder what the true odds are at that point of getting a fourth one.
Your (0.0475)^3 value for 3 consecutive blackjacks is for wampling with replacement; my calculation is for sampling w/o replacement, which I believe to be appropriate unless either (1) the first BJ occurred on the last hand of the shoe, or, (2) penetration is one hand per shoe. In the first instance the probability would be based on sampling w repl for the first two hands, followed by sampling w/o repl for the third hand. In the second instance, do not play unless the game otherwise has positive basic strategy expectation.
We never got an answer on whether tied naturals count or not, so the whole thing is just an approximation. Note that it hardly matters; we're talking about pennies per hand.
Don
on the previous shoe(nt).
