Incidentally, your original calculation shows the danger of rounding intermediate values:
So we have a 1.44% chance of that happening. 1.44796
Gives us 1.18% for that second scenario. 1.15837
So 2.62% of my having 15 and the dealer showing a 10 or face card. 2.60633
How do you do a quick monte carlo simulation? Is there a software specific for that? I often want to run one but writing the program usually discourages me.
I get 2.61%.
960 possibilities if your first card is a five (4 x 16 x 15)
960 possibilities if your first card is worth ten (16 x 4 x 15)
6 x 256 possibilities if your first card is A, 4, 6-9 (4 x 4 x 16)
= 3456 out of a total of (52 x 51 x 50) or 132,600 possibilities
= 2.61%
I think an infinite # of decks would change that to 2.54%, since the dealers third card would not be affected by your ten, and the number of possibilities would be 52 x 52 x 52 since cards aren’t used up.
What are the chances, in blackjack, that I will start with 15, and the dealer will be showing a 10 (or face card worth 10)?
Despite the fact that my buddy thinks the odds of this are near 100%, I’ve tried to calculate the actual odds (assuming a single deck).
I’ve come up with 2.62% -ish. Not really sure if I’m right.
I started with all the possible hands I could have.
2652 (52 * 51)
Then I figured out how many hands would give me 15.
64 hands with a 10 (or equivalent) and a 5.
64 hands with a 5 and a 10.
16 hands with a 9 and a 6.
16 hands with a 6 and a 9.
16 hands with an 8 and a 7.
16 hands with a 7 and an 8.
16 hands with an ace and a 4.
16 hands with a 4 and an ace.
These fall into two groups, those that have a 10, and therefore deprive the dealer of one, and those that don’t.
128 hands have a 10.
96 hands are 15 some other way.
If I have a 10, she has 15 cards left, out of 50, to get a 10.
If I don’t she has 16 out of 50.
So my odds of making a 15 using a 10 (or face card) are 4.8%.
Her odds of then having a 10 showing are 30%.
So we have a 1.44% chance of that happening.
-Plus-
Me making 15 some other way: 3.7%.
And her having a 10 showing: 32%.
Gives us 1.18% for that second scenario.
So 2.62% of my having 15 and the dealer showing a 10 or face card.
Two questions.
Assuming a single deck, is this correct?
Assuming multiple decks, what changes (if anything)?
If I’m wrong, where did I go wrong? (It would help my brain to have a combination of english and math to explain where I went wrong, rather than just something like “you should have used a factorial for possible hands 52!-4!”, etc…)
Thanks.
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