Blackjack Bonus at San Pablo Card Room

The card room inside the bingo slot parlor in San Pablo, CA (20 min outside of San Francisco) has that California Blackjack where a natural is 22 so it complies with the no regular blackjack law. Anyway, they have a "broken hearts bonus" that pays $100,000 if you are dealt three 8 of clubs, you bust (so you would have to not split the 8's which many would not do) and the dealer gets three 7 of hearts. They use a six deck shoe that is NOT on constant shuffle. I play a couple bucks here every couple of weeks and I ask the dealers if they have ever seen anyone get it, they have all said no. Also there is another bonus that pays $25,000 that is basically the same thing but a different combination of suits (I think 8 of spades and 7 of diamonds). I was curious to the probability of hitting this. Other than the obvious practically impossible, perhaps with a small bet and receiving a pair of 8 of clubs and the dealer showing a 7 of hearts it might be worth the risk to hit instead of split. Any thoughts on this? I believe it does not require a side bet to qualify for this.

Assume 6 decks (312 cards)

Probability of getting two 8 of clubs: 6/312 * 5/311.
Probability of dealer getting 7 of hearts given you have two 8 of clubs: 6/310

So probability of getting correct initial hand: 6*5*6/(312*311*310) = 0.0000059841

Probability of getting a 3rd 8 of clubs given initial hand: 4/309

Probability of dealer drawing two 7 of hearts given all above: 5/308*4/307

Total probability: 6*5*6*4*5*4 / (312*311*310*309*308*307) = 0.0000000000163847

So... very very small. 1 in 61 billion. You could win the lottery a few times before hitting this one.

In fact, the expectation of a $100'000 bonus is only $0.000001638, which is a lot less than the cost of hitting a pair of 8s against a 7 versus the correct play of splitting. So... this bonus is actually a trap to get you to make a playing mistake if you are dealt the initial hand.

Thank you for figuring it out! How funny would it be if you were playing, lets say $5 a hand and did not hit and the dealer could have ended up with the cards! Since no side bet is required to get this bonus, it is one of those things that are likely to never happen but are fun to look at.

Quote: dwheatley

Assume 6 decks (312 cards)

Probability of getting two 8 of clubs: 6/312 * 5/311.
Probability of dealer getting 7 of hearts given you have two 8 of clubs: 6/310

So probability of getting correct initial hand: 6*5*6/(312*311*310) = 0.0000059841

Probability of getting a 3rd 8 of clubs given initial hand: 4/309

Probability of dealer drawing two 7 of hearts given all above: 5/308*4/307

Total probability: 6*5*6*4*5*4 / (312*311*310*309*308*307) = 0.0000000000163847

So... very very small. 1 in 61 billion. You could win the lottery a few times before hitting this one.

In fact, the expectation of a $100'000 bonus is only $0.000001638, which is a lot less than the cost of hitting a pair of 8s against a 7 versus the correct play of splitting. So... this bonus is actually a trap to get you to make a playing mistake if you are dealt the initial hand.

Good analysis. That has got to be the worst 'promotion' I have ever seen. I hope people ignore it.