Now, how could the Count pass up a chance to talk about that most numerically significant time of year, Tax Time!? With lots of talk about rising tax rates and socialism this year, I thought it might be fun and informative to take a look at tax tables from previous years and compare them to recent times.

Using a spreadsheet I found on the internet1, I created a table of tax burdens by year for different income levels. While the tax brackets have seen highs in the 90s percent in the post-war 40s and 50s, and started with lows of 1-7% in the early 1910s, I figured 1970 was a good starting point for modern relevancy. I further pared this information down to just the years after fairly significant changes to the tax codes, as many of the annual lines were very similar. I also chose to use the Married Filing Singly tax table, as it was most often the same as Single, and almost always half the income amounts of Married Filing Jointly for the same tax. Here’s is the resulting data:

• The percentage tax burden isn’t the tax “bracket”, it’s the total amount paid in taxes if your income was at the displayed level. Of course, tax shelters exist, but this is based on “taxable income”, the number you use when you look up how much tax you owe.
• Quite obviously, the 1970s were not a good time to be making money. Yes, home interest rates were in the teens, so anyone who owned a home had quite a nice write-off in the interest they were paying, but good Lord, that’s some high taxes!
• 1988 was a good year to make a lot of money, where people making $100k actually paid a (slightly) higher percentage of their income in taxes than those making 3 or even 5 times as much!

Finally, I notice that only at 1993 do we see a marked rise in taxes, and only for those making $100k or more; it seems that taxes have done nothing but drop since 1970 except for that one year. Of course, things like the Alternative Minimum Tax and changes to Capital Gains taxes change the landscape, making it very difficult to get a clear picture of true tax liabilities.

This is quite interesting! People are never happy paying taxes, but I think it’s refreshing to see we’ve got it quite lucky nowadays compared to our parents, tax-wise. Now, if only we could get that money spent just the way we want it, no? But that’s another topic. I sense a post about unusual budget items! Look for it in the future; right now, I’ve got to go file my taxes.

The Count

1. Historical US Tax Tables

Interested in the data I compiled for the above chart? Here it is:

What?! Another birthday so soon!? No, you haven’t lost 6 months of your life, and The Count isn’t 41… yet! Due to the overwhelming popularity of my Birthday article, it became clear to me that even those people that weren’t born during the awesome year of 1969 would like to be able to see their own age in huge-numbered detail.


After many hours of toil, The Count has made available to you, his loyal reader, a page designed to give you all the joy he felt in seeing his age in Earth years converted to galactically-large numbers of other units. Simply click the link below, and enjoy!

The Count

Have you ever had a problem stuck in your head, and you couldn’t find the answer? I was recently reminded of a problem I first came up with while doing a statistics workbook the summer of my 3rd grade year (yes, my math-teacher mother gave us workbooks to do during summer break… hey, it got results). The book dealt with dice and the probabilities of a 1 showing on a 6-sider, or the sum of 2 rolled dice being 7, etc. but my question had a twist I couldn’t quite solve. Now it’s easy to see that the probability of rolling a 1 on a 6-sided die is 1 in 6, but what probability exists, in rolling 2 dice, of seeing at least one 1 (on either die, or both)?


So there I was, 7 years old and stuck with a math problem I couldn’t solve. Well, what would you do? That’s right, I asked people. Over the next 9 years, I asked math teachers and other adults I thought might be able to help, but no one seemed able to explain it to me. Of course, I could brute force the answer for 2 or even 3 dice quite easily as well (and did), but by that time I wasn’t interested in just the single answer to my problem, but a more general solution for N dice. The difficulty of the problem comes from the “at least one” phrasing. Using discrete math, you could end up having to compute the probability for each of the numbers of dice smaller than your requested count and use alternating subtraction and addition to account for subset solutions and overlap.

Of course, I did come up with (or was given) attempts at a quicker solution. They usually fell into 2 types. The first type was the simplest; since we were rolling two dice, and the probability with 1 die was 1 in 6, the new probability must be 2 in 6. Unfortunately, this easily extrapolates to say that when rolling 6 dice, you absolutely must get a 1, 2, 3, 4, 5, and 6. Anyone who plays Yahtzee can certainly see the flaw there. The second type of false solution came from, at some point, someone recalling some remnant of a college statistics course (the part where previous rolls shouldn’t affect subsequent rolls) and concluding that the answer must be still 1 in 6. This answer also fails to satisfy quickly when you consider more than just 2 dice being rolled. How could the probability of getting at least one 1 when rolling 10 or even 100 dice still be just 1 in 6?

It wasn’t until I reached college and took an actual course in statistics that I found the answer. Fortunately, there’s an easy method. If you ever find a statistics problem that uses the “at least one” phrase, the best bet is to turn it around. What are the odds of not getting any 1s on those dice? As it happens, that’s merely 5 in 6 for each die rolled, multiplied together. In my problem, with only 2 dice, the probability of not getting a 1 is 5/6 * 5/6 = or 25 in 36. Now, since all the other possible results must contain a 1, we’re left with a solution of (36-25) = 11 in 36! This method works for N dice as well, with the probability of getting at least one 1 out of N dice rolled being:

I’m a football fan… well, more specifically the NFL. I hear that colleges other than the one I went to actually have their own teams and that quite a lot of people think that the outcomes of these “college football games” are important, but I never saw the appeal. Anyway, as I said, I follow the NFL. This year, as the season has proceeded, I’ve found myself noticing quite a few long winning and losing streaks occuring this year. With 2 teams reaching 10-0, and a few teams having streaks of 6 and 7 wins (or losses), I decided to track the actual numbers and determine just how abnormal this season is.

Using one of the many available NFL statistic sites, I was able to compile the number of streaks that occured at each of the 10 lengths available at week 11 (which was just completed). Note that only 10 games have been played by each team at this point, and that I ignored the bye week, allowing streaks to continue through the bye uninterrupted. The following are my results:

Happy Birthday to me. Yes, that’s right, you’re friendly Count was born 40 years ago today, which just happened to be Labor Day in 1969. In light of this special event, I’ve prepared a little set of numbers just for fun. None of this is particularly important, just a bit of mind-candy for my birthday. So let’s go!

40 years on this Earth has some easy conversions into smaller time periods. 480 months makes 2,080 weeks pretty easily. Days is more of a problem, as there are some irregularities in the Leap Day counts. As it happens, those 40 years turned into 14,600 days at 365/year plus 10 leap days is 14,610 total days. This computes to 350,400 hours, times 60 for 21,024,000 minutes, and again for 1,261,440,000 seconds. That’s right, over a billion seconds I’ve been alive!


Just because I lived all those 40 years on Earth, doesn’t mean that time can’t be measured by the years of the other planets in our system. Check out how long my life has been on those planets!1

I’ve done a fair bit of travelling in my life; as an Army brat, you’re sent across the country or across the world every 3 or 4 years. Then again, merely moving around on the face of this planet is chump change compared to how far I (and you) have really travelled during my life. Just sitting on the surface of Earth means I travelled around 40,076 km/day (the planetary circumference2), for a total of 585,109,600 km travelled in 14,600 Earth-sized circles. With an orbital path of 240,800,000 km/year3, I’ve also travelled 9,632,000,000 km (1/10th of 1 percent of a light year) in larger circles around the Sun. Finally, the Sun’s moves along its orbital path in the galaxy at about 251 km/s. At this rate, I’ve traveled over 316.6 billion km, over 1/30th of a light year through the universe during my life… so far.

Hey, this was fun! I hope you’ve enjoyed my numerical recounting (get it?) of my life. I feel a lot better about just how small a number 40 is. Maybe I’m not so old, after all.

The Count

1. Convert Earth years to other planets
2. Earth on Wikipedia
3. Orbital lengths
4. The Sun on Wikipedia

• The number of games played per year in MLB has changed over the years but, thanks to other helpful fans1, I’m told the number of games played since 1900 is 174,206, roughly 1600/year. Of course, each game involves two teams.
• While not technically a minimum (due to shortened games), a normal game would require at least 51-54 batters (depending on whether the bottom of the 9th is played). I’m not unhappy using 54 as a valid minimum, giving us 9,407,124 plate appearances during those games… at least!
• While games with fewer than 100 pitches per team aren’t all that rare, nor are they common. Yet games with 120+ pitches for a team happen all the time. At only 100 pitches per team per game on average, that’s 34,841,200 pitches thrown during those games. I believe that the real number of pitches thrown is much higher, but it gives you an idea of the scale of numbers we’re talking about.

Now, let look at some of the rarified statistics of the game:

• 206 times a single pitcher has thrown a no-hitter (like Mr. Sanchez)2, pitching a whole game where no opposing batter get a hit. That’s once every 3400 games or so (or every 6800 starting pitchers).
• 245 times a player has “hit for the cycle”3, hitting at least a single, double, triple and home run during a single game. That’s around 2 per season since 1900, or 1 every 40000 plate appearances!
• 15 (now 16, read on) times a single pitcher has thrown a perfect game4, a complete game where no opposing player reaches base. That’s once every 11,600+ games, or once every 7 years or so. The last was over 5 years ago, but the longest time period between perfect games was 34 years from 1922-1956, so don’t get your hopes up. edit: Of course, this isn’t true anymore, as a new perfect game was thrown on July 23rd, 2009 by Mark Beuhrle7
  • 10 times, the pitcher allowed the first batter of the game on base, but then threw a perfect 27 (or more!) outs.
  • 9 times a perfect game has been lost to the last (27th) batsman, who reached base.
  • 8 times a perfect game has been lost when the only batter to reach base did so on an error, just like Mr. Sanchez’ game. Only once was it the pitcher himself who committed the error.
  • Once, a pitcher named Harvey Haddix threw 12 perfect innings (36 batters) before an error ruined his perfect game, he even took the loss when that batter scored to end the game.
• 13 times a batter has hit 4 home runs in the same game. That’s once every 13,400 games. Mike Cameron hit four solo home runs (no one else was on base), and Bob Horner’s team actually lost the game where he hit his 4 dingers.

These are certainly rare occurrences, but by no means the rarest. As I said, baseball is a game of numbers, and you can find any number of special sets of circumstances that fans have only had the chance to see once or maybe twice in the last 100+ years. From hitting 2 grand slams in an inning (once, by Fernando Tatis) to pitching 21 strikeouts in a game (once, by Tom Cheney), these are records worthy of the hard-core baseball fan. Having the chance to talk about the wonderful numbers that result from one pitch, one at-bat, one inning, game, or season is some of the best fun of being a baseball fan. I hope you enjoy it as much as I do.


The Count

1. Games played in MLB since 1900 on Yahoo! Answers
2. No hitters on Wikipedia
3. Hit for the cycle on Wikipedia
4. Perfect game on Wikipedia
5. 4 home-runs in one game by Baseball Almanac
6. Baseball single-game records on Wikipedia
7. Buehrle Perfect Game article on White Sox Traditions