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Ask the Wizard #171
Jack from Troy
You are on the right track with the binary numbers but that is not quite the winning strategy. First, if you can leave your opponent with an odd number of rows of one each then do so. Otherwise break down each row into its binary components. For example, 99 would be 64+32+2+1. Then add up the number of each component over all the rows. Then look for a play that will leave your opponent with an even number of all binary components over all the rows.
Let’s look at an example. Suppose it is your turn with the following scenario.
The following table breaks down each row into its binary components.
Player’s Turn 1
| Row | 1 | 2 | 4 | 8 | 16 |
| 6 | 0 | 1 | 1 | 0 | 0 |
| 9 | 1 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 1 | 0 | 0 |
| 5 | 1 | 0 | 1 | 0 | 0 |
| 25 | 1 | 0 | 0 | 1 | 1 |
| Total | 3 | 1 | 3 | 2 | 1 |
You can see that there is an odd number of ones, twos, fours, and sixteens. Clearly we need to get the row of 25 under 16 to eliminate the 16 unit. To keep the total of the binary components even we need to remove the 1, add a 2, add a 4, keep the 8, and remove the 16. That means the best play is 2+4+8=14 in the last row. Leaving 14 in the bottom row we have the following.
Computer’s Turn 1
| Row | 1 | 2 | 4 | 8 | 16 |
| 6 | 0 | 1 | 1 | 0 | 0 |
| 9 | 1 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 1 | 0 | 0 |
| 5 | 1 | 0 | 1 | 0 | 0 |
| 14 | 0 | 1 | 1 | 1 | 0 |
| Total | 2 | 2 | 4 | 2 | 0 |
The computer takes its turn, leaving us with this.
Here is the binary breakdown of that.
Player’s Turn 2
| Row | 1 | 2 | 4 | 8 | 16 |
| 6 | 0 | 1 | 1 | 0 | 0 |
| 9 | 1 | 0 | 0 | 1 | 0 |
| 2 | 0 | 1 | 0 | 0 | 0 |
| 5 | 1 | 0 | 1 | 0 | 0 |
| 14 | 0 | 1 | 1 | 1 | 0 |
| Total | 2 | 3 | 3 | 2 | 0 |
Here we need to remove a 2 and a 4, to get those totals even. There is only one row, the 14, which has both components. So remove 6 from that, leaving 8.
Computer’s Turn 2
| Row | 1 | 2 | 4 | 8 | 16 |
| 6 | 0 | 1 | 1 | 0 | 0 |
| 9 | 1 | 0 | 0 | 1 | 0 |
| 2 | 0 | 1 | 0 | 0 | 0 |
| 5 | 1 | 0 | 1 | 0 | 0 |
| 8 | 0 | 0 | 0 | 1 | 0 |
| Total | 2 | 2 | 2 | 2 | 0 |
The computer takes its turn, leaving us with this.
Now we need to change the 1, 4, and 8 columns.
Player’s Turn 3
| Row | 1 | 2 | 4 | 8 | 16 |
| 6 | 0 | 1 | 1 | 0 | 0 |
| 4 | 0 | 0 | 1 | 0 | 0 |
| 2 | 0 | 1 | 0 | 0 | 0 |
| 5 | 1 | 0 | 1 | 0 | 0 |
| 8 | 0 | 0 | 0 | 1 | 0 |
| Total | 1 | 2 | 3 | 1 | 0 |
That can be done by changing the row of 8 to 5 as follows.
Computer’s Turn 3
| Row | 1 | 2 | 4 | 8 | 16 |
| 6 | 0 | 1 | 1 | 0 | 0 |
| 4 | 0 | 0 | 1 | 0 | 0 |
| 2 | 0 | 1 | 0 | 0 | 0 |
| 5 | 1 | 0 | 1 | 0 | 0 |
| 5 | 1 | 0 | 1 | 0 | 0 |
| Total | 2 | 2 | 4 | 0 | 0 |
The computer takes its turn, leaving us with this.
Now we need to change the 2 and 4 totals.
Player’s Turn 4
| Row | 1 | 2 | 4 | 8 | 16 |
| 6 | 0 | 1 | 1 | 0 | 0 |
| 4 | 0 | 0 | 1 | 0 | 0 |
| 2 | 0 | 1 | 0 | 0 | 0 |
| 5 | 1 | 0 | 1 | 0 | 0 |
| 3 | 1 | 1 | 0 | 0 | 0 |
| Total | 2 | 3 | 3 | 0 | 0 |
This can be done by changing the 6 to a 0.
Computer’s Turn 4
| Row | 1 | 2 | 4 | 8 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 1 | 0 | 0 |
| 2 | 0 | 1 | 0 | 0 | 0 |
| 5 | 1 | 0 | 1 | 0 | 0 |
| 3 | 1 | 1 | 0 | 0 | 0 |
| Total | 2 | 2 | 2 | 0 | 0 |
The computer takes its turn, leaving us with this.
Now we need to change the 2s and 4s.
Player’s Turn 5
| Row | 1 | 2 | 4 | 8 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 1 | 0 | 0 | 0 |
| 2 | 0 | 1 | 0 | 0 | 0 |
| 5 | 1 | 0 | 1 | 0 | 0 |
| 3 | 1 | 1 | 0 | 0 | 0 |
| Total | 2 | 3 | 1 | 0 | 0 |
This can be accomplished by changing the row of 5 to 3. If you can ever get your opponent to an x,x,y,y situation you can’t help but win, if you can maintain the same situation until the end.
Computer’s Turn 5
| Row | 1 | 2 | 4 | 8 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 1 | 0 | 0 | 0 |
| 2 | 0 | 1 | 0 | 0 | 0 |
| 3 | 1 | 1 | 0 | 0 | 0 |
| 3 | 1 | 1 | 0 | 0 | 0 |
| Total | 2 | 4 | 0 | 0 | 0 |
The next few moves I keep the computer on x,x,y,y patterns. Here the computer leaves me with 2,2,3,2; so I leave it with 2,2,2,2.
The computer then gives me 2,2,1,2. I leave it with 2,2,1,1.
The computer then leaves me with 2,2,1. I leave it with 2,2. If you can ever get your opponent to two equal rows you can’t help but win, just keep the rows equal.
The computer then leaves me with a single pile of 2, and I remove 1.
Here is the end of the game.
Jay from Whitestone
I agree. I don’t recommend making put or place bets, because as you said, the odds are better making a line bet and then maximum odds. However, some people absolutely must bet on the points directly. If that is a given, I advocate making the best bet between the place, buy, and put, which I explain in greater depth in my craps section.
Brett from Alhambra
My blackjack appendix 9 was created for just these kinds of questions. Assuming 6 decks, my table shows the expected value of doubling 8+2 vs. 7 is 0.396342. The expected value of doubling 8+3 vs. 10 is 0.176919. So you are right, 10 vs. 7 is the much better hand to have.
Kevin from Vernon, NJ
Thanks! Your way is double counting getting two of the 20 cards you need. The probability you get one of your 20 needed cards on the turn is 20/47 = 0.4255. The probability you don’t get it on the turn and then do get it on the river is (27/47)*(20/46) = 0.2498. So the total probability is 0.4255 + 0.2498 = 0.6753.
I recently witnessed a situation at the local tribal casino involving protocol at the craps table that puzzled me, and I'm wondering if you clarify it for me. On his come out roll, Player ’A’ threw a number and established a point. For the next roll, the player next to ’A’ (Player ’B’) picked up the dice and shot. It’s not clear why he did this, or if he even knew ’A’. One of the other players objected, pointing out that ’B’ was not the original shooter. After much discussion and head scratching by the dealers and the boxman, the dice were passed to the next player (Player ’C’) who finished the hand (he eventually sevened out).
Was this the correct protocol for this situation, and if so, what is the logic behind it? If Player ’A’ for some reason simply didn't want to shoot any more, why shouldn’t he be allowed to relinquish the dice? If Player ’B’ picked up the dice because he didn't understand the game, or if the stickman mistakenly put them in front of him, shouldn't the dice go back to ’A’ to finish the hand?
David from Bainbridge Island, WA
I asked the Bone Man at nextshooter.com this one. Here is what he said.
If a player other than the correct shooter picks up the dice and rolls them, it should be a no-call, no-roll and the dice should be returned to the correct shooter. Though this is indeed the proper ruling, the boxman in some instances may allow for the roll if the result is to the favor of all or most of the players. In some instances, the result of the roll may not effect any of the player(s) wagers. Also... Any player can request that the dice be sent to the next shooter to finish a hand. In such cases the same dice may be sent out or the new shooter can request new dice. Upon the completion of the roll, the SAME shooter can then shoot HIS/HER OWN HAND, thereby having more than one hand.
Terje from Stockholm
I started to use the Normal approximation to solve this, but the probability of over 100 points is too low for that method to be accurate. So I did a random simulation of 8.25 million trials and the number of trails that were 101 points or more was 127. So the probability is about 1 in 65,000.
Jason from Vancouver
When the prizes become life-changing amounts, the wise player should play conservatively at the expense of maximizing expected value. A good strategy should be to maximize expected happiness. A good function to measure happiness I think is the log of your total wealth. Let’s take a person with existing wealth of $100,000 who is presented with two cases of $0.01 and $1,000,000. By taking “no deal” the expected happiness is 0.5*log($100,000.01) + 0.5*log($1,100,000) = 5.520696. Let b be the bank offer where the player is indifferent to taking it.
log(b) = 5.520696
b = 105.520696
b = $331,662.50.
So this hypothetical player should be indifferent at a bank offer of $331,662.50. The lesser your wealth going into the game the more conservatively you should play. Usually in the late stages of the game the bank offers are close to expected value, sometimes a little more bit more. The only rational case where a player could win the million is if he had a lot of wealth going into the game and/or the bank offers were unusually stingy. The producers seem to like hard-working middle class people, so we’re unlikely to see somebody who can afford to be cavalier when large amounts are involved. I have also never seen the bank make offers under 90% of expected value late in the game. The time when we will see somebody win the million is when a degenerate gambler gets on the show who can’t stop. When that happens I will be rooting for the banker.
Milton from Santa Fe
Thanks. To answer this question you first have to ask yourself why you are gambling in the first place. If you are trying to lose as little as possible then you shouldn’t play at all. However, if you are playing for the fun of gambling then I would choose the $10 game with the 0.2% edge. The expected loss will be the same but you’ll get more of a fix with the larger wagers.
William from Mississauga, Ontario
The break even point is $102,680.24. I just added an analysis of this bet to my section on Pai Gow Poker side bets.
Mike from Philadelphia, PA
I agree that a "2/4" game means the bets are in $2 units before the turn and $4 after the turn. However, the small blind is likely be $1. It sounds like when Arizona Charlie's is calling a game "2/4" they are referring to the blinds, which would mean a small blind of $2, and a big blind of $4, which would imply $8 bets after the turn. I've never heard this usage before so I don’t blame you for being surprised.
p.s. I later received the following from Anthony, a poker room supervisor.
I am a poker room supervisor. I was just writing to let you know about the correct wording for the poker limits in Texas Hold 'em. If as game is referred to as $2/4. It is referring to the bets. (A game is only referred to by the blinds if it is No-Limit.) $2 pre flop and post flop,$4 on the turn and the river.A game that is listed as $2/$4/$8 (which is exactly what I suspect the previous letter writer was actually referring to, is $2 pre-flop,$4 post-flop,$8 turn and river .It is also possible to have a game listed as $2/$4/$6/$8. Just passing the information on, good poker supervisors want people to have as much information as possible.
