How to calculate % of busting in Blackjack?

Question

How do you calculate the probability that a dealer will bust based on their up card in BlackJack, for example when the dealer's up card is 2 then the percentage of them busting is 35.3%. How do you get that?

From Player Advantage vs. Dealer Up Card chart

http://www.blackjacktactics.com/blackjack/odds/

Answer 1

It is a lengthy but simple interative process of determining how often every possible outcome will occur.

Assuming we are using a single 52-card deck, we know that the value of the dealers up card is "2" and that there are 51 other cards in the deck. Of those cards,

4 out of 51 are an ace
16 out of 51 are a 10, J, Q, or K
4 out of 51 are a 9
4 out of 51 are an 8
4 out of 51 are a 7
4 out of 51 are a 6
4 out of 51 are a 5
4 out of 51 are a four
4 out of 51 are a three, and
3 out of 51 are a two.

For each of those possibilities, you calculate what the odds are for the next card. For example, if the hole card was a 2, of the 50 cards remaining

4 out of 50 are an ace
16 out of 50 are a 10, J, Q, or K
4 out of 50 are a 9
4 out of 50 are an 8
4 out of 50 are a 7
4 out of 50 are a 6
4 out of 50 are a 5
4 out of 50 are a four
4 out of 50 are a three, and
2 out of 50 are a two.

You continue this process until you get a hand where the dealer would stand of bust. So, for example, the odds of the dealer ending up busting with a 2 up and a 2 in the hole, a 9 on the first hit to his hand and another 9 on the second hit to his hand would be

3/51 * 4/50 * 3/49

Once you have calculated the odds of every possible outcome you can simply sum the odds of all the bust hands to get the odds of the dealer busting with a 2 up.

While this would be a long and tedious process to do by hand, a simple computer program can calculate the odds in less than a second.