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Last Updated: November 6, 2016
Blackjack House Edge With Infinite Decks
Mike: Hi, this is Mike with the Wizard of Odds. In my last video, I showed you how you can create the Blackjack Basic Strategy starting with just the blank spreadsheet, which is in front of me right now. And in this second part, I'm going to show you, how to go from here to getting a house advantage in Blackjack. So I already know the expected amount the player's going to win or lose for any hand.
Now all I need to do is find the probability of any starting hand and then multiply those probabilities by the expected values and finally make an adjustment for the dealer having a blackjack to begin with.
So let's start with a page, let's call it Prob for probability. This will stand for the probability of every possible starting hand. So the player could start with anything from a five up to a 21 or a blackjack. He could have a soft anything from a 13 to a blackjack. So with the hard totals, we will put the blackjacks with the soft totals. And there's the possible split. So we'll say a pair from two up to ten and then the Aces. And of course, we do everything according to the dealer's up card. All right.
The player's first card could be a two through an Ace, as well as the second card. So let's find how often each possible sum occurs. So not counting the Aces. This table shows the total for every combination of the first card and the second card. I'm going to get rid of the pairs because I treat those separately because the player can split those. All right. So here's all the possible hard totals and in this column I'm going to indicate the probability of each one as seen by how often they occur in this table up here. So I'm going to use the sum if function, which I use all the time. Wait, it's little premature to do that. Okay. Here's a similar table.
I'm going to make another table that shows the probability of each total. So in most cases, it's just one in 13 times, one in 13, when there's no ten involved. When there's a ten involved, it's one in 13 times, four in 13.
Okay. Now we're already. Equal sum if, and we're searching through this table for this number and when we find it, we sum up these numbers. And I have to put in the dollar sign, so when I copy and paste this, the range will stay the same of the two matrices.
Let me make this bigger for you guys because you probably can't see it very well. Sorry about that. Okay.
So here we've got the probability of each player hard total from five up to 19. So up here, I'm pretty much doing the same thing that across the dealer total. But you don't need to bother. So I can simply say for the hard five for example, as refer to my table down here and then times one in 13 for the dealer up card of the two. And I'm going to take away this 20 because the idea with that with the pairs. Okay.
For a dealer up card to ten, I do the same thing but I multiply by four divided by 13 because there's four, ten point cards in the deck but now I'm also going to multiply by 12 and 13 because we know that the hole card is not an Ace. And a similar thing with the Aces, except we're back to multiplying by one 13th, for the dealer Ace and nine and divided by 13, for the fact that the hole card is not a ten point card. So here's all the probabilities of each starting hand for player hard totals. So according to this 67.6% of the time, the player will be dealing with a hard total. Okay.
The soft totals with most of them, it's going to be two times, one in 13 squared. And the reason for the two is because with a soft 13 for example, the two cards, the Ace and the two could be an either, or, so you multiply by two. And then also multiply by one in 13 for the dealer's up card. So the total probabilities two times, one in 13 queued. And this is going to work all the way through the soft 20. With the soft 21, there's a greater chance of that because there's more ten point cards than any of the other cards. So we're going to do two times, one in 13 squared, times four in 13. Okay.
So I copied and pasted that, all the way through the dealer's nine. Actually I copied and pasted too far. So now let's do the ten. Again, we're going to change one of the one in 13's to a four in 13 and also multiplied by 12 and 13, because the dealer doesn't have an Ace in the hole. For soft 13 again it's an Ace. We can keep it the same, two times, one in 13 queued and then times nine and 13 because we know the dealer doesn't have a ten point card in the hole. Okay.
The probability of a player blackjack against the dealer ten, is two times one in 13 times four and 13 times four and 13, times 12 and divided by 13. Probability of a player blackjack divided by- I mean against a dealer Ace is two times, one in 13 times, four and 13 times, one in 13 times, nine and 13. Hopefully that's right.
Now, let's deal with the pairs. Probability of a player pair of twos, against the dealer two, is one in 13 queued. And we don't multiply by two because we're dealing with two of the same card. That's going to work all the way through the dealer nines, as well as the pair of Aces, for a pair of tens against the dealer two, we do four and 13 squared times one in 13, for the dealer two. Copy and paste that down. Okay.
Now let's do the same thing but for the dealer ten up. That's going to be one in 13 squared, times four and 13, times 12 and 13. The probability of a player paired tens against the dealer ten, is four and 13 queued, times 12 and 13. All right.
Probability of a pair of twos against a dealer Ace, is one of 13 queued, times nine in 13. Probability of a player paired tens against the dealer Ace is four in 13 squared, times one in 13 times nine in 13. Okay.
There's all the probabilities, hopefully they're right. In fact, let's add them up and see if they are. Okay.
That adds up to 95.27%. And the part we're missing is the dealer blackjack. The probability of the dealer Blackjack is two times four and 13 times one in 13 or 4.74%. So let's add up these and hopefully they add up to one. They do, great. So there's a probability sheet.
Next, let's make a sheet for the Expected Returns.
The next step is going to be to create an expected return sheet which we'll call ER for short. This is going to contain the expected return of any given hand, which we've already figured out. We're just going to summarize it all in one convenient sheet, in the same kind of layout as the probability sheet. So for a hard five against a two, we just need to refer to the Hit, Stand, Double, Surrender sheet. Copy and paste that down. Same thing with the soft totals. And the pairs. But here for the pairs, we're going to refer to the splitting sheet from the last video.
Next, let's make an Expected Value sheet which we'll call EV. This is going to be the probability of the expected value and the probability for any given hand. So we simply need to multiply the probability sheet, any cell in the probability sheet by the corresponding cell in the expected to turn sheet. And do that for every possible starting hand. And let's see what it all adds up to. A positive 1.43%. But as I've been saying all along, we've been assuming the whole way that the dealer does not have a blackjack. And I just realized, I made an error, in the Expected Return sheet for a soft 21, I've been treating it like an Ace five and a five. When actually it should be 1.5 because it's a blackjack. So let me copy and paste this 1.5 all the way down, that changes the Expected Return to 4.02%. But again that's- once you clear the hurdle of no dealer blackjack, what's the probability of a dealer blackjack? That is two times, four in 13 times, one in 13 and the two is because the ten and Ace could be in either order.
So there's also a possibility that there is a winning dealer blackjack. And the probability of that is the probability of a blackjack, times the probability that the player does not have a blackjack. And we're going to multiply by negative one because the player loses in that situation. There's also the possibility of a blackjack tie but that results in a push. So that's not...so we would be adding and subtracting zero, so we don't need to bother with that. So what's the grand total? Let's call this, No Dealer Blackjack. Just the sum of these two cells and the answer is negative 48.5%. So there you have it. That is my expected return for a blackjack game with infinite decks, dealer stands on a soft 17, double after a split is allowed, split only one time. Surrender allowed, resplitting Ace is not allowed. If you look up these rules but under an eight deck game, you'll get .43%. The difference between .43 and .485 is due to that infinite deck assumption. So there you have it. The House Edge in blackjack assuming an infinite deck, again just starting from nothing.
Thank you and hope you enjoyed it.