Jul 04, 2007 Is there an easy way to memorize the blackjack basic strategy chart? I don't really want to have the chart in front of me at the casino, so I'm wondering if anyone has any hints about how to easily memorize the blackjack basic strategy chart? Dec 26, 2015 Blackjack has the lowest house edge over the player of all games ever. So, in essence, it is the most player-friendly game in the world (Best ways to win).Of course, you also have to know what you doing (a basic glossary can be found here): there is a large volume of rules that not many people know but that make a difference between a winner and a loser.
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Michael Shackleford: Hi, this is Mike with the Wizard of Odds. The purpose of this video is to create the blackjack basic strategy from scratch in Excel starting with just a blank spreadsheet.
Why am I doing this? Because I'm getting a little tired of hearing dealers say that the basic strategy is a myth, that it was created by Ed Thorp, Beat the Dealer fame back in 1962 on an old UNIVAC computer or whatever and he must've made a mistake somewhere and this mistake has been repeated for decades by other gambling writers who never once thought to confirm his work and we've all just been repeating his work like parrots.
Number one I'd like to correct something is actually, Ed Thorp was not the first person to publish the basic strategy. There were four guys in the army in the 50s who did it before him and published it in the American Statistical Journal. However, Ed Thorp was the first one to write a book about it including addressing card counting and broke all kinds of new ground in the game.
I might also add that blackjack analysis has been repeated over and over by lots of gambling writers, including myself, and we all-- all the legitimate writers basically agree with each other.
Nevertheless, dealers will continue to say, 'Well, I've seen players play by the book and lose lots of times.' Well, of course, it's a game of luck and bad players who don't follow the basic strategy lose, too. But, as I always say, the more ridiculous a belief is, the more tenaciously it tends to be held, but let's quit talking and actually show you the basic strategy starting from nothing.
I'm going to use an assumption of an infinite number of decks of cards for the sake of simplicity. The mathematics of the game are not tremendously different between eight decks and an infinite number of decks. The basic strategy will be different in a couple borderline plays as we'll see later. But let's jump in, shall we?
Let's call the first sheet in the spreadsheet, 'the dealer sheet.' This is going to show the probability of every possible outcome for the dealer according to all possible 10 up cards.
So, one of these things are going to happen to the dealer; he could bust, he could end up with 17, 18, 19, 20, or 21, and we're going to do this by American rules where the dealer peeks for blackjack and he exposes it, if he doesn't-- if he has it otherwise, the player is not drawing dead to a handy can't be. We will factor in the blackjacks later on in another video where I show you how to calculate the house advantage.
There's 10 possible up cards the dealer can have, which we'll put across the first row. This table up here will represent the possible probabilities under the American rules. Again, it's a little tricky because in the-- again, because of this peeking rule. So, let's do it underneath here according to the European rule where the dealer doesn't peek for a blackjack and let's consider all the possible totals all the way up to 31 which will make-- doing the spreadsheet a little bit easier and we also will look at the soft totals and we'll start there from a 12, also up to a 31. What's a soft 31, you might ask? That's the same thing as a hard 21. For a soft 22 or more, just subtract 10 and it's the same thing as a hard total.
All right, if the dealer has a-- that's the 17. There's a 100% chance he will end up with a 17 because he's going to stop and same thing with an 18 through a 21. The dealer ends up with a 22 or more. There's a 100% chance he's going to bust.
With the soft hands, let's assume the rule that the dealer stands on a soft 17. We can do the same thing there. With a soft 22, that's going to be the same thing as a hard 12.
Okay, let's start with looking at what is the probability of the dealer getting a 17 if he starts with a two? Well, the next card could bring him up to a four all the way up to an 11, each with equal chance. There's 4 chances he could end up with a 12 because there's four ranks in the deck that are 10 points, and one chance he could get to a soft 13. We divide by 13 because there's 13 ranks. This same formula is going to work all the way up to 16 for all the other possible totals.
Now, let's work on the soft hands. The dealer starts with-- hit's a soft 12. He could end up with a 13 up to a 21 each with equal chance, and four chances at a 22, copying paste that down.
Let's take the total to make sure it adds up to one. Good, it does. Let's make sure it does for all the other outcomes. Looks good so far.
All right, now, let's think about the American rules. Blackjack is the same, all other things being equal in the US and Europe if the dealer's showing a two through nine because there's no chance a dealer can have a blackjack, so we can just refer to the European numbers for two through nine.
Let me add another row here for the total. For the 10 and the ace, there's no chance the dealer has a blackjack because he already peeked for it and again, we'll adjust for that later on when we figure out the house edge. But if the dealer has a 10 showing and when he hits that, that could leave him with a 12 up through a 19, each with one chance and four chances of the 20, and we divide it by 12 because there's 12 ranks, not counting the ace. Similar kind of logic with the ace, but there you can end up with a soft 12 up to a soft 20 and divide by 9, because there's only nine ranks that won't give them a blackjack.
There are all your dealer probabilities. Let's save this before I forget. We'll just call this 'basic strategy.' Now, we're ready to start talking about the player. Let's make our next sheet called 'stand.' This is going to show the players expected value of standing for any possible scenario.
Let's do everything by the 10-possible dealer up cards. Let's do all the player sheets according to having anywhere from a hard four all the way to 31 and a soft 12 all the way to 31.
Let me make the screen a little bigger for you. I hope that helps so you can see what I'm doing. All right. What if the dealer stands on a four against a two? The only way he's going to win is if the dealer busts. His expected value is the probability of the dealer busting minus the probability of anything else happening.
So, he can expect to lose by standing on a four against a two of about 29.3% of his bet. That's the same number for standing on everything all the way through a 16 because a 16 is no better than a four or a zero.
Things start to get better with a 17 because now, a 17 is going to turn from a loss to a push, so the expected value there is the probability of the dealer busting minus the probability of an 18 through a 21. We can ignore the dealer getting a 17 because that's going to result in a push with a player. Eighteen, expected value is the probability of the bust or 17 minus the probability of a 19 to21.
With a 19, expected value is the probability of a dealer having a bust 17 or 18 minus a 20 or 21. With a 20, it's a bust through a 19 minus a 21 and with a 21, it's anything except another 21 which would be a push.
So, the expected value of a 17 against a two is a expected loss of about 15.3% of the bet; with an 18, it's positive 12.2; with a 19, positive 38.6; with a 20, positive 64.0% and with a 21, positive 80.2%. If the dealer gets to 22 or more, these loss's expected value is negative one.
So, let's copy and paste that all over through to an ace. Now, with the soft totals, it doesn't help the player if it's soft if he's just going to stand on it. We can just refer to the hard totals. Soft 22 is the same thing as a soft 12 and expected value for standing. There's our expected values of if the player stands.
Let's talk about hitting next. We'll make a sheet called 'hit' and let's also make one called 'HS' which will represent the expected value of the better choice between hitting and standing because the player should make the choice with the higher expected value at every stage of the game. Let's zoom this so you can see it better.
Okay. For the hit stand sheet, at any given stage of the game, the expected value is going to be the higher of the expected value for that state between hitting and standing. We're going to need to reference that sheet when we talk about hitting because after the player hits, he can still keep hitting.
So, if the player hits a hard four against the two for example, he might be left with anywhere from a six through a 13, each with equal chance, four chances at a 14 and one chance at a soft 15 and divide it by 13 and just copy and paste.
The player hits a 21, he's going to bust, so his expected value is negative one, or if the player hits a soft 12, well, then he can have anywhere from a soft 13 to 21, four chances at the soft 22 which is the same thing as a 12, and again, divide by 13. Copy and paste that down through 21. As usual, a soft 22 is the same thing as a 12, so we can refer to the hard totals there.
So, there's our expected values for hitting. Now, let's see what the basic strategy looks like if the player is only limited to hitting and standing which is the case with three or more cards. If this greater expected value is the same as standing, let's put an S, otherwise an H for hitting.
Now, let's color that in. Blackjack writers seem to always use red for hitting, I think it comes from Lawrence Revere, and yellow for standing. Here's what it looks like. Take it from me, this is correct. Any legitimate blackjack book will show you this is the correct strategy for hitting and standing for any possible situation.
Let's talk about doubling next. Let's make a new sheet called 'double.' Let's make this bigger for you again. The doubling sheet is going to be based on the stand sheet, because when you double, you get one card only. There's no option to hit after that.
So, if the player doubles on a four against the two, he could end up with a six through a 13 or four chances at a 14 or a soft 15 and we divide it by 13. Let's multiply all this by two because now, the player asked to double his bet. We can simply copy that down through the 11. With the 12, it gets a little bit different because now the ace is going to count as-- it must count as a one.
So, we just adjust the formula a little bit like that. There are all your expected values for doubling for hard totals. Now, let's do it for the soft totals, start out by with the two because again, the player's doubling his bet.
If the player starts with a soft 12, his next card could give him anywhere from a soft 13 through 21, and four chances at a soft 22, which, again, is the same thing as a hard 12 and divided by 13. A soft 22 is again the same thing as a hard 12, so there's all your expected values for doubling.
Now, let's make a sheet. We'll call it 'HSD' for Hit Stand Double, which is going to represent the expected value of the best choice between hitting, standing, and doubling. So, this is going to be the maximum of hitting, standing, and doubling.
Let's adjust this little sheet here, 'the basic strategy,' to show the best choice between hitting, standing, and doubling. Voila, and let's color all this in-- color in all the doubling squares blue because everyone seems to use blue for doubling.
Here's our basic strategy for doubling. Again, take it from me, you'll see this same chart in any legitimate blackjack book, and here it is for the soft totals. Now, here are the couple differences between four to eight decks and infinite decks. These are two very borderline situations.
With four to eight decks, you're also going to double a soft 15 against a four and you're going to double a soft 13 against a five. Again, the only reason these are different is because of the infinite decks. There, we have doubling.
Next, let's talk about surrendering. Surrendering is really easy. Let's call this 'SUR' for surrender. The expected value of surrendering-- sorry, let me make this bigger for you again. It keeps changing. The expected value of surrendering is negative 0.5 for any situation where the player hasn't busted yet, because the player, by definition of the game, is losing half his bet and he doesn't have to play his cards out.
Let's make another sheet. We'll call this one 'HSDR,' that's going to represent the maximum expected value between hitting, standing, doubling, and surrendering. Why do we use an R for surrender? Because S was already taken for standing. Why does this zoom keep changing?
All right. So, the expected value between all those four choices is the maximum of hitting, standing, doubling, and-- oops. Surrendering. You can see right here, it's only optimal to surrender if the player has a 16 and the dealer is showing a 9, 10, or ace, or a 15 against the dealer 10. Let's update our chart here. I hope my wife gets that. I guess she's not. I hope it wasn't too important. There you see the four surrenders that I just set.
Finally, let's talk about splitting. The player could have a pair of anything from a two through 10s or two aces. Okay, what's the expected value if the player splits a pair of twos against a two? Well, first, let's put a two because the player has a double as a bet, and after splitting that, each hand could be anything. After he hits each one, it could be anywhere from a four through an 11, four chances at the 12, and one chance at a soft 13.
By the way, I am assuming that the player may only split once, no re-splitting, just to keep things simple. We copy and paste that. Things are different with aces. With aces, we're going to go to the stand sheet because the way blackjack is played is you only get one card after splitting aces.
So, after splitting an ace with each ace, you could have it anywhere from a 12 through 20 and four chances of the 21. Here's your expected values for splitting. Well, let's see when those are better than the best alternative.
Down here, let's look at the better value between, for example, with a pair of twos against the two, the higher of the splitting expected value or a hard four. With a three, it's going to be greater of the expected value of splitting the threes or a hard six, so, we're going to have to alternate the rows in the standing sheet.
For the pair of twos, we looked at cell B2 in the sheet for hitting, standing, or doubling. For a pair of threes, we're going to look at cell B4; for the fours, D6; for the fives, B8; and so on.
Okay. With aces, it's going to be the greater of splitting the aces or playing it out as a soft 12. Copy and paste it for the other dealer up cards. Now, let's make a little chart to show when it's best to split.
So, if the best expected value equals the splitting expected value, we'll put a Y for 'Yes, you should split,' otherwise put a, 'No.' Copy and paste that down and let's color it in.
If it's equal to a Y for 'Yes, we should split,' let's use green. Here, we have the splitting basic strategy. Take it from me, look in any blackjack book and it's going to tell you that you should split in these situations with four or more decks or 48 decks and the dealer standing on a soft 17.
So, there's your basic strategy. Here's this basic strategy for splitting. Here's a basic strategy for all your hard totals and your soft totals. Again, all that starting from just a blank spreadsheet. It wasn't all that complicated. I hope you understood this. I'm going to put this spreadsheet in Google Documents so you can look at it more closely on your own if you want to.
In my next video, I'm going to expand on this one to show you how we go from all these expected values to the actual house advantage of the game.
Thank you and I hope that wasn't too boring.
Michael Shackleford: Hi, this is Mike with the Wizard of Odds. The purpose of this video is to create the blackjack basic strategy from scratch in Excel starting with just a blank spreadsheet.
Why am I doing this? Because I'm getting a little tired of hearing dealers say that the basic strategy is a myth, that it was created by Ed Thorp, Beat the Dealer fame back in 1962 on an old UNIVAC computer or whatever and he must've made a mistake somewhere and this mistake has been repeated for decades by other gambling writers who never once thought to confirm his work and we've all just been repeating his work like parrots.
Number one I'd like to correct something is actually, Ed Thorp was not the first person to publish the basic strategy. There were four guys in the army in the 50s who did it before him and published it in the American Statistical Journal. However, Ed Thorp was the first one to write a book about it including addressing card counting and broke all kinds of new ground in the game.
I might also add that blackjack analysis has been repeated over and over by lots of gambling writers, including myself, and we all-- all the legitimate writers basically agree with each other.
Nevertheless, dealers will continue to say, 'Well, I've seen players play by the book and lose lots of times.' Well, of course, it's a game of luck and bad players who don't follow the basic strategy lose, too. But, as I always say, the more ridiculous a belief is, the more tenaciously it tends to be held, but let's quit talking and actually show you the basic strategy starting from nothing.
I'm going to use an assumption of an infinite number of decks of cards for the sake of simplicity. The mathematics of the game are not tremendously different between eight decks and an infinite number of decks. The basic strategy will be different in a couple borderline plays as we'll see later. But let's jump in, shall we?
Let's call the first sheet in the spreadsheet, 'the dealer sheet.' This is going to show the probability of every possible outcome for the dealer according to all possible 10 up cards.
So, one of these things are going to happen to the dealer; he could bust, he could end up with 17, 18, 19, 20, or 21, and we're going to do this by American rules where the dealer peeks for blackjack and he exposes it, if he doesn't-- if he has it otherwise, the player is not drawing dead to a handy can't be. We will factor in the blackjacks later on in another video where I show you how to calculate the house advantage.
There's 10 possible up cards the dealer can have, which we'll put across the first row. This table up here will represent the possible probabilities under the American rules. Again, it's a little tricky because in the-- again, because of this peeking rule. So, let's do it underneath here according to the European rule where the dealer doesn't peek for a blackjack and let's consider all the possible totals all the way up to 31 which will make-- doing the spreadsheet a little bit easier and we also will look at the soft totals and we'll start there from a 12, also up to a 31. What's a soft 31, you might ask? That's the same thing as a hard 21. For a soft 22 or more, just subtract 10 and it's the same thing as a hard total.
All right, if the dealer has a-- that's the 17. There's a 100% chance he will end up with a 17 because he's going to stop and same thing with an 18 through a 21. The dealer ends up with a 22 or more. There's a 100% chance he's going to bust.
With the soft hands, let's assume the rule that the dealer stands on a soft 17. We can do the same thing there. With a soft 22, that's going to be the same thing as a hard 12.
Okay, let's start with looking at what is the probability of the dealer getting a 17 if he starts with a two? Well, the next card could bring him up to a four all the way up to an 11, each with equal chance. There's 4 chances he could end up with a 12 because there's four ranks in the deck that are 10 points, and one chance he could get to a soft 13. We divide by 13 because there's 13 ranks. This same formula is going to work all the way up to 16 for all the other possible totals.
Now, let's work on the soft hands. The dealer starts with-- hit's a soft 12. He could end up with a 13 up to a 21 each with equal chance, and four chances at a 22, copying paste that down.
Let's take the total to make sure it adds up to one. Good, it does. Let's make sure it does for all the other outcomes. Looks good so far.
All right, now, let's think about the American rules. Blackjack is the same, all other things being equal in the US and Europe if the dealer's showing a two through nine because there's no chance a dealer can have a blackjack, so we can just refer to the European numbers for two through nine.
Let me add another row here for the total. For the 10 and the ace, there's no chance the dealer has a blackjack because he already peeked for it and again, we'll adjust for that later on when we figure out the house edge. But if the dealer has a 10 showing and when he hits that, that could leave him with a 12 up through a 19, each with one chance and four chances of the 20, and we divide it by 12 because there's 12 ranks, not counting the ace. Similar kind of logic with the ace, but there you can end up with a soft 12 up to a soft 20 and divide by 9, because there's only nine ranks that won't give them a blackjack.
There are all your dealer probabilities. Let's save this before I forget. We'll just call this 'basic strategy.' Now, we're ready to start talking about the player. Let's make our next sheet called 'stand.' This is going to show the players expected value of standing for any possible scenario.
Let's do everything by the 10-possible dealer up cards. Let's do all the player sheets according to having anywhere from a hard four all the way to 31 and a soft 12 all the way to 31.
Let me make the screen a little bigger for you. I hope that helps so you can see what I'm doing. All right. What if the dealer stands on a four against a two? The only way he's going to win is if the dealer busts. His expected value is the probability of the dealer busting minus the probability of anything else happening.
So, he can expect to lose by standing on a four against a two of about 29.3% of his bet. That's the same number for standing on everything all the way through a 16 because a 16 is no better than a four or a zero.
Things start to get better with a 17 because now, a 17 is going to turn from a loss to a push, so the expected value there is the probability of the dealer busting minus the probability of an 18 through a 21. We can ignore the dealer getting a 17 because that's going to result in a push with a player. Eighteen, expected value is the probability of the bust or 17 minus the probability of a 19 to21.
With a 19, expected value is the probability of a dealer having a bust 17 or 18 minus a 20 or 21. With a 20, it's a bust through a 19 minus a 21 and with a 21, it's anything except another 21 which would be a push.
So, the expected value of a 17 against a two is a expected loss of about 15.3% of the bet; with an 18, it's positive 12.2; with a 19, positive 38.6; with a 20, positive 64.0% and with a 21, positive 80.2%. If the dealer gets to 22 or more, these loss's expected value is negative one.
So, let's copy and paste that all over through to an ace. Now, with the soft totals, it doesn't help the player if it's soft if he's just going to stand on it. We can just refer to the hard totals. Soft 22 is the same thing as a soft 12 and expected value for standing. There's our expected values of if the player stands.
Blackjack How To Deal
Let's talk about hitting next. We'll make a sheet called 'hit' and let's also make one called 'HS' which will represent the expected value of the better choice between hitting and standing because the player should make the choice with the higher expected value at every stage of the game. Let's zoom this so you can see it better.
Okay. For the hit stand sheet, at any given stage of the game, the expected value is going to be the higher of the expected value for that state between hitting and standing. We're going to need to reference that sheet when we talk about hitting because after the player hits, he can still keep hitting.
So, if the player hits a hard four against the two for example, he might be left with anywhere from a six through a 13, each with equal chance, four chances at a 14 and one chance at a soft 15 and divide it by 13 and just copy and paste.
The player hits a 21, he's going to bust, so his expected value is negative one, or if the player hits a soft 12, well, then he can have anywhere from a soft 13 to 21, four chances at the soft 22 which is the same thing as a 12, and again, divide by 13. Copy and paste that down through 21. As usual, a soft 22 is the same thing as a 12, so we can refer to the hard totals there.
So, there's our expected values for hitting. Now, let's see what the basic strategy looks like if the player is only limited to hitting and standing which is the case with three or more cards. If this greater expected value is the same as standing, let's put an S, otherwise an H for hitting.
Now, let's color that in. Blackjack writers seem to always use red for hitting, I think it comes from Lawrence Revere, and yellow for standing. Here's what it looks like. Take it from me, this is correct. Any legitimate blackjack book will show you this is the correct strategy for hitting and standing for any possible situation.
Let's talk about doubling next. Let's make a new sheet called 'double.' Let's make this bigger for you again. The doubling sheet is going to be based on the stand sheet, because when you double, you get one card only. There's no option to hit after that.
So, if the player doubles on a four against the two, he could end up with a six through a 13 or four chances at a 14 or a soft 15 and we divide it by 13. Let's multiply all this by two because now, the player asked to double his bet. We can simply copy that down through the 11. With the 12, it gets a little bit different because now the ace is going to count as-- it must count as a one.
So, we just adjust the formula a little bit like that. There are all your expected values for doubling for hard totals. Now, let's do it for the soft totals, start out by with the two because again, the player's doubling his bet.
If the player starts with a soft 12, his next card could give him anywhere from a soft 13 through 21, and four chances at a soft 22, which, again, is the same thing as a hard 12 and divided by 13. A soft 22 is again the same thing as a hard 12, so there's all your expected values for doubling.
Now, let's make a sheet. We'll call it 'HSD' for Hit Stand Double, which is going to represent the expected value of the best choice between hitting, standing, and doubling. So, this is going to be the maximum of hitting, standing, and doubling.
Let's adjust this little sheet here, 'the basic strategy,' to show the best choice between hitting, standing, and doubling. Voila, and let's color all this in-- color in all the doubling squares blue because everyone seems to use blue for doubling.
Here's our basic strategy for doubling. Again, take it from me, you'll see this same chart in any legitimate blackjack book, and here it is for the soft totals. Now, here are the couple differences between four to eight decks and infinite decks. These are two very borderline situations.
With four to eight decks, you're also going to double a soft 15 against a four and you're going to double a soft 13 against a five. Again, the only reason these are different is because of the infinite decks. There, we have doubling.
Next, let's talk about surrendering. Surrendering is really easy. Let's call this 'SUR' for surrender. The expected value of surrendering-- sorry, let me make this bigger for you again. It keeps changing. The expected value of surrendering is negative 0.5 for any situation where the player hasn't busted yet, because the player, by definition of the game, is losing half his bet and he doesn't have to play his cards out.
Let's make another sheet. We'll call this one 'HSDR,' that's going to represent the maximum expected value between hitting, standing, doubling, and surrendering. Why do we use an R for surrender? Because S was already taken for standing. Why does this zoom keep changing?
All right. So, the expected value between all those four choices is the maximum of hitting, standing, doubling, and-- oops. Surrendering. You can see right here, it's only optimal to surrender if the player has a 16 and the dealer is showing a 9, 10, or ace, or a 15 against the dealer 10. Let's update our chart here. I hope my wife gets that. I guess she's not. I hope it wasn't too important. There you see the four surrenders that I just set.
Finally, let's talk about splitting. The player could have a pair of anything from a two through 10s or two aces. Okay, what's the expected value if the player splits a pair of twos against a two? Well, first, let's put a two because the player has a double as a bet, and after splitting that, each hand could be anything. After he hits each one, it could be anywhere from a four through an 11, four chances at the 12, and one chance at a soft 13.
By the way, I am assuming that the player may only split once, no re-splitting, just to keep things simple. We copy and paste that. Things are different with aces. With aces, we're going to go to the stand sheet because the way blackjack is played is you only get one card after splitting aces.
So, after splitting an ace with each ace, you could have it anywhere from a 12 through 20 and four chances of the 21. Here's your expected values for splitting. Well, let's see when those are better than the best alternative.
Down here, let's look at the better value between, for example, with a pair of twos against the two, the higher of the splitting expected value or a hard four. With a three, it's going to be greater of the expected value of splitting the threes or a hard six, so, we're going to have to alternate the rows in the standing sheet.
For the pair of twos, we looked at cell B2 in the sheet for hitting, standing, or doubling. For a pair of threes, we're going to look at cell B4; for the fours, D6; for the fives, B8; and so on.
Okay. With aces, it's going to be the greater of splitting the aces or playing it out as a soft 12. Copy and paste it for the other dealer up cards. Now, let's make a little chart to show when it's best to split.
So, if the best expected value equals the splitting expected value, we'll put a Y for 'Yes, you should split,' otherwise put a, 'No.' Copy and paste that down and let's color it in.
If it's equal to a Y for 'Yes, we should split,' let's use green. Here, we have the splitting basic strategy. Take it from me, look in any blackjack book and it's going to tell you that you should split in these situations with four or more decks or 48 decks and the dealer standing on a soft 17.
So, there's your basic strategy. Here's this basic strategy for splitting. Here's a basic strategy for all your hard totals and your soft totals. Again, all that starting from just a blank spreadsheet. It wasn't all that complicated. I hope you understood this. I'm going to put this spreadsheet in Google Documents so you can look at it more closely on your own if you want to.
In my next video, I'm going to expand on this one to show you how we go from all these expected values to the actual house advantage of the game.
How To Memorize Blackjack Basic Strategy
Thank you and I hope that wasn't too boring.