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Ask the Wizard #219

If all thirty major league baseball teams were of equal talent, then what would be the probability that any given team from a specified division would make the playoffs? Clearly the current rules favor the National League Central and disadvantage the American League West.

Why don’t teams in the American League West complain about this inequity? The differences here to me are not trivial. Since making the playoffs is big money for a team, I’m surprised that there aren’t more complaints from the NL Central. I would also be a little miffed, as a fan of any of these six teams, to know my team is getting the short end of the stick.

I can’t imagine that I’m the first one to ever notice this? Does MLB compensate these teams in some way?

Ben

For the benefit of other readers, there are two leagues, each with three divisions, in Major League Baseball. Each division has five teams, except the American League West, with 4, and the National League Central, with 6. Every year, in both leagues, the three division leaders, and a Wild Card team, make the playoffs. The Wild Card team is the team with the best win/loss record in the League, not counting the three division leaders. There are some tie-breaking rules, which I won’t get into, and assume they are resolved randomly.

Indeed, the American League West has a big advantage, and the National League Central has a big disadvantage, all other things being equal. I do not know of any compensating rules. Nor do I know the reason for this imbalance. Before 1998 there were only two divisions. In 1998, Major League Baseball added two new teams, the Tampa Bay Devil Rays and Arizona Diamondbacks. They also increased the number of divisions from four to six, and added the Wild Card rule. However, why they didn’t balance the leagues, I have no idea. The best solution to this inequity, in my opinion, would be to move the Houston Astros to the American League West. Some may say Houston is not west enough, but the Texas Rangers are also in that division.

I posted the answer and solution to the probability question at my companion site, mathproblems.info, as problem number 200.

p.s. Since I posted this column, one reader wrote that the reason for the imbalance was to keep the number of teams in each league an even number. This allows every team to play on a given day, and keep the play within the division. However, this I don’t buy as an excuse. In 2008 the regular season consisted of 162 games per team, played over 185 days (not counting the all-star game day, and a day on each side). So, each team played 0.8757 games per day. Of those 162 games, 18 are played against teams in the opposite division, and 144 in the same division. I suggest that with balanced divisions of 15 teams each, on any given day 12 teams play within their own league. Over 185 days, that will accomplish 185 × (12/15) = 148 games. In the other 37 days, schedule 14 interleague games, for a total of 162 games. So, the only change will be a reduction in the number of interleague games per team from 18 to 14. It seems to me, most fans oppose interleague games to begin with, including me.

p.p.s. Another reader wrote to say that my system would not accommodate the baseball traditions of keeping every team playing on Saturdays and Sundays, and designating interleague play to only certain times of the seasons. Okay, fair points. However, if tradition is so important in baseball, why introduce interleague games at all? Personally, I value fairness over tradition. Put me in charge of baseball scheduling, and I’ll no only balance the leagues, but keep every team playing on weekends. However, it would come at the expense of the days off being clumped together. Maybe the easier thing to do would be to add two more teams. My hometown of Las Vegas will be the first to volunteer to be one of them.

I have a question about a series of bets in craps. The strategy is called the "Iron Cross." It involves a bet on the 5, 6, 8, and the field. I read up on this, and found that this particular bet will pay on every roll that is not a 7. I was told that this gives you the lowest house edge. What are all the various odds and what-nots to go along with it?

Carter from Calgary

If the player bets $5 on the field and 5, and $6 on the 6 and 8, then he will have a net win of $2 on the 5, 6, and 8, $10 on the 2, $15 on the 12, and $5 on the other field numbers, assuming that the 12 pays 3 to 1 on the field. The player will lose $22 on a 7. On a per roll basis, the player can expect to lose 25 cents compared to $22 in bets, for a house edge of 1.136%.

This begs the question, why is this lower than the individual house edge of each bet made? It’s not. The reason it seems that way is the result of comparing apples to oranges. The house edge of place bets is usually expressed as the expected loss per bet resolved. Looking at the individual bets on a per-roll basis, the house edge on the 5 is 1.11%, and on the 6 and 8 is 0.46%, according to my craps appendix 2. Comparing apples to apples, the house edge is a weighted average of the house edge on the field, 5, 6, and 8, on a per-roll basis, or (5/22)×2.778% + (5/22)×1.111% + (6/22)×0.463% + (6/22)×0.463% = 1.136%.

Do the casinos put oxygen in the air inside the casino, so that you will not get tired?

Alisha from Pontotoc

No. This is just an urban legend.

Is it true that state lottery jackpot annuities cease to pay if you die before the annuity runs its term? I heard this was true of New York. This seems outrageous to me.

Alex from Montreal

I checked the New York and California lottery web sites. Both indicated that if the winner dies before all the payments are made, the rest will be paid to the winner's designated heir or estate.

Hello, I am a seventh grader from Hawaii. I am doing a science fair project on poker and shuffling. I am using your charts about the win percentage of pocket Texas Hold ’Em hands, according to the number of players in the game. I was hoping you could answer a few questions that would help me with my project:

  1. How did you come up with the percentages found in the charts?
  2. If you used a computer program, how did you develop it and how long did it take?
  3. You stated that you started the Wizard of Odds as a hobby. Did experimenting change as your site became more well-known? Why or why not?

    Anonymous

    1. The two-player table was done by a brute-force looping program, that cycled through all 1225 possible opponent cards, and 1,712,304 possible community cards. For three to eight players, looping would have taken a prohibitive amount of time, so I did a random simulation.
    2. I write almost all my programs in C++, including both programs I just mentioned. The rest are in Java or PERL. I mostly copied and pasted code from other poker-based programs. The new code only look about a day to write.
    3. Yes, I started my site as a hobby in June 1997. It wasn’t until January 2000 that I accepted advertising, and tried to make a business out of it. It has gone through three different domains over the years. Here is what it looked like in May 1999. The purpose of the site has always remained the same, a resource for mathematically-based gambling strategy. Through the years, I have just been adding more games and material. One experiment was providing my NFL picks for the 2005 season, which was an abject failure.