A look at probability of winning based on expected return

I'm a newbie and I have questions regarding some numbers I put together in this excel spreadsheet in google docs. I left the document in excel format because it uses features that the Google Docs format can not handle. I apologize in advance if I have made any rookie mistakes or errors. I did not see this topic discussed yet. I'm hoping my work is correct and that this leads to some good conversation.

This spreadsheet does a couple of things.

First, it uses the Expected Return (ER) tables from the wizard's blackjack appendix 1 to automatically reproduce the best play strategy that is presented in that same appendix. This is the top left chart (note that the chart colors are automatically generated using conditional formatting.)

The second thing the spreadsheet does is to calculate the probability of winning each possible play. For standing and
hitting plays I calculated the probability of winning using (ER + 1)/2. I got this formula using the following logic
where W equals the probability of winning and L equals the probability of losing:

ER = ($1 * W) - ($1 * L)
ER = (1 * W) - (1 * (1 - W)) // W + L = 1, so L = 1 - W
ER = 2W - 1
2W = ER + 1
W = (ER + 1)/2

For Doubling and Splitting plays, I calculated the probability of winning using (ER + 2)/4 obtain from:

ER = ($2 * W) - ($2 * L)
ER = (2 * W) - (2 * (1 - W))
ER = 4W - 2
4W = ER + 2
W = (ER + 2)/4

As you would expect, the calculated probabilities for winning a hand is less than 50% for negative ERs, and greater than 50% for positive ERs. This leads me to believe that these formulas are correct.

The second colored chart (middle top) in the spreadsheet automatically determines the best play strategy considering only the probability of winning the hand. Again, the colors are automatically generated. One thing to note is that the chart is set up to favor Hitting over Doubling or Splitting if there is no difference in the probability of winning between the different plays.

The third colored chart was put together by hand and shows the advantage in the probability of winning the hand if the second chart (based on probability of winning) is used instead of the standard best play chart based on expected return. Note that I did not create all this to argue against using best play strategy based on ER. I just wanted to understand how Doubling and Splitting affected the win/loss result for any one particular hand.

The first thing you will notice about the best play chart based on probability of winning is that there is no Doubling. This result makes perfect sense to me. You can never increase your chances of winning a hand by limiting yourself to one additional card. In the situations where you would take only one more card anyhow (ex. 11 vs 4,5, or 6) the probability of winning the hand stays the same. For the other Double Down situations, you risk getting stuck with a non-optimal hand (for example, you Double a 11 vs. 7 and pull a 2 for a total of 13.) I'm hoping that this spreadsheet simply confirms that.

Some of the results for Split possibilities seem rather interesting. For example, it seems you can increase your chances of winning a hand by not Splitting 9,9 vs. 2 thru 6. It also appears that you can increase your chances of winning a 7,7 vs. 8 by almost 9% by Splitting (if all my logic is correct).


Questions:

1. Are the formulas above for calculating probability of winning based on ER correct?

2. Does the programming in the spreadsheet look sound?

3. I understand that play based on ER is the best if you are betting the same amount every hand. But, I wonder if these results show that it may make sense to alter play if you also alter the size of your bets. For example, if someone has a very large bet on the table (relative to their base bet), should they consider passing on all Double Down situations since doubling can not increase their chances of winning the hand? Does the answer depend on the magnitude difference between that big bet and the base bet?

Quote: johnnyc

The first thing you will notice about the best play chart based on probability of winning is that there is no Doubling. This result makes perfect sense to me. You can never increase your chances of winning a hand by limiting yourself to one additional card. In the situations where you would take only one more card anyhow (ex. 11 vs 4,5, or 6) the probability of winning the hand stays the same. For the other Double Down situations, you risk getting stuck with a non-optimal hand (for example, you Double a 11 vs. 7 and pull a 2 for a total of 13.) I'm hoping that this spreadsheet simply confirms that.

Doubling on a 10 vs. dealer 4-6 would not decrease your chances of winning the hand either. Remember that when doubling down on 10 (or less) the ace is a high card (unlike doubling on 11 in which the ace is low). With both doubling on 10 and 11 your final total will always be somewhere between 12 and 21 inclusive (doubling on 11 is a bit more favorable though since a 10-value card would give you 21 rather than 20).

Sorry, but I goofed up again by clicking the "quote" button instead of the "edit" one.

Quote: Kellynbnf

Doubling on a 10 vs. dealer 4-6 would not decrease your chances of winning the hand either.

Yes, my spreadsheet clearly shows this. If you look at the third, far right, colored chart you will see 0.00% advantage in winning percentage by Hitting 10 vs 4-6 as opposed to Doubling.

I'm going to try and simplify my own post in an attempt to get more interest. See my original post above if you want all the details. I am not trying to make an argument that betting systems work.

If you make the same bet every time, you will win more money by Doubling Down in the situations where it is called for by basic strategy. However, since you are limited to only one additional card when you Double Down, you can never increase your chances of winning a particular hand by Doubling.

If the situation is one where you would only take one additional card anyhow (10 or 11 vs 4-6) your chances of winning the hand do not change by Doubling. However, for the other Double Down situations, you can end up stuck with a non-optimal hand total when you Double Down. An example of this would be Doubling an 11 vs. 7 and pulling a 2 for a total of 13. You would certainly take another hit in this situation if you could. Doubling Down can never increase your chances of winning a hand, and it may decrease your chances of winning a hand.

Let's say someone starts a blackjack session with $500 at a $5 table and they (wrongly) have faith in a Martingale system and plan to double their bet after every loss ($10, $20, $40, $80, etc.) until they run out of money or win a hand. They plan to make a $5 bet after every win.

When this person is making a bet for an amount other than $5, meaning they are risking more money to recover their previous loss (or loses), wouldn't it be advantageous for them to play more conservatively and pass on Double Down situations in order to limit their risk and, in some situations, increase their chances of winning that particular hand?

You are suggesting that there might be situations where the probability of winning the hand might be a more important success criteria than the expected return. I disagree with this. At the end of the day, I am only concerned with the amount of chips that I color up. (And how much fun I had, and how many free drinks I got!) I don't care about how many hands I won.

I especially disagree with your premise of passing on double/split opportunities when you have extra money on the table. If I were varying my bet for whatever reason, and I only got one double/split opportunity during an hour of play, I'd want it to occur when I have my maximum bet on the table so I can take the greatest advantage.

Quote: PapaChubby

I especially disagree with your premise of passing on double/split opportunities when you have extra money on the table.

Thanks for replying. I can't argue with your position on this.

Many years ago I took a gambling class in Chicago. He taught a progressive betting technique (1-2-2-3-4-6-8-etc units.) For his playing strategy, he had situations where he would Double Down only if his bet was 1 or 2 units. I put together this spreadsheet (link in original post) to try to see if there was any math behind his suggestion. I believe he was trying to increase his chances of winning the hand, and keeping his winning streak going, when he was higher in his betting progression. I am not promoting this type of betting.

I could see someone using this type of play with a Martingale approach as well. Believing that with bigger bets it is better to be less greedy and play for a win (and avoid ruin) instead of playing to increase winnings.

Insurance is another example of my question. Say you are normally a $10 better and you have a $1000 bet on the table. You get 21, and the dealer has an Ace showing. Would you use standard basic strategy (never take insurance) in this situation or would you take even money?

If I bet $1000 every hand, in the long run, never taking insurance is the best play. But, for the above situation, I'm not sure that it is a bad move to take even money, since $1000 is 100 times greater than the base bet of $10.

An instance where the probability of winning a hand might be more important than the overall EV is in certain instances of tournament play. For example, if your original bet is enough to reach your goal and double that amount won't help you any more (or can even hurt you in some cases if you lose). Conversely, sometimes in tournament play you might sometimes double even in cases where it's strongly supotimal under Basic Strategy if you need to reach a certain goal beyond what you initally bet.

Just a quick observation on the spreadsheet - shouldn't the expected return of standing on A,10 (aka blackjack) against a dealer 2-9 be 1.5? And the corresponding win percentage should be 100%, no?

If you varied your play in the way suggested you would be decreasing your overall EV at the cost of a higher chance of winning the hand. You should only do this if you in a risk of ruin situation, or in a tournament

Quote: UWPeteO

Just a quick observation on the spreadsheet - shouldn't the expected return of standing on A,10 (aka blackjack) against a dealer 2-9 be 1.5? And the corresponding win percentage should be 100%, no?

All the expected return numbers come from the Wizard's blackjack appendix 1. Looking at his own spreadsheet for calculating those numbers, I believe he is considering 21 to be a "hand total" and not a two card 21 in that situation. Since you would never hit a 21, it doesn't affect best play strategy.