It is official. I am a nerd. This may have been suspected before, but now it is official. I spent 30 minutes of my boat voyage from Mwanza to Bukoba (nice voyage, by the way) trying to figure out the probability that the dealer in blackjack would go bust if his first card is a 6. This is in a game where he has to hit on 16, and has to stop on 17. I didn't do the whole thing because it turns into too many scenarios once you get up to 5 or more cards to make the bust. But, with just 3 cards, the prob of a bust is (32 + 7 + 6 + 5 + 4)/(169) = 54/169, which is approximately 4/13, or slightly less than 1/3. If we allow the dealer to go bust with 4 cards, then we only add a small amount to the probability of the bust, and the bust probability is still slightly less than 1/3. At this point, even I got bored with the project and decided not to calculate the scenarios of going bust on 5, 6, or 7 cards. However, these are very small probabilities and would not add much. Therefore, the total probability of going bust, when the dealer's first card is a 6, in only approximately 1/3.
Why is this crucially important, so important that I had to work on the problem on the boat, rather than look at the water? Remember the guy last night who kept advising his friends about when to hit and when to stay in blackjack? Well, one of his cardinal rules, as I remember, was that, if the dealer has a 6 showing, and you can't go bust with one more card, then you should ALWAYS double up. Double up means that you double your bet and you get one more card. I remember thinking at the time that the rule seemed a bit faulty, but I couldn't calculate the whole thing there, while sitting at the table.
However, I now think I've shown that this is not a good rule. For example, if your 2 cards are a 3 and a 2, then, no matter what, your third card will give you a score of 16 or less. This means that, no matter what, the dealer has a 2/3 chance of beating you! You should not double up on that. Thank you. Crucially important. Of course, I now feel slightly like that kid in that novel entitled something-something-curious-incident-something, where the kid does stuff like counts the spots on cows, and he solves some fancy maths problem. He wants to put the solution as a chapter in his book, but his teacher convinces him to just make it an appendix, on the grounds that not too many people would be interested. I now suspect that this entire entry should have been an appendix to my blog, but that's life.
When we arrived in Bukoba, on Wednesday morning, it was very beautiful, and here is the view.
Here is a picture of our boat, taken from the upper deck, as we approached Bukoba.
2 comments:
I think you are confuded about the double up rule...
What happens is you split your cards, place the same bet on the second card as the first then get to play both hands one at a time as if you were tow players. Taking as many cards as you want. So the reason for wanting to double up is the odds are fair the dealer will go bust and even if they do not bust then the odds are fair they will not get as good a hand as you can get starting with hands that do not have a 6 in them.
Hi Anonymous,
You are describing a split, not a double up. In a double up, you get exactly one more card. For example, if you have 5-5 to start with, you could either: (a) split and get as many cards as you want for each 5, or (b) double up, and get one more card to add to your current score of 10. In this case, of course, you would never split since 10 is a good starting hand, but you would often want to double up (unless, maybe, the dealer starts with an A).
So, I was describing the double up situation, since that's the one where that other guy at my table suggested that you should *always* do it if the dealer starts with a 6.
Anyway, don't think this is super interesting for most of the blog readers - but apparently my mystery anonymous reader would have read the appendix had it been made an appendix! That's nice to know!
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