Posts tagged ‘baseball’
Today is Opening Day in Major League Baseball, and 13 games will be played today.
It’s also Square Root Day, because the date 4/4/16 transforms to 4 × 4 = 16.
With those two things in mind, here’s a trivia question that seems appropriate. Identify the data set used to create the graph below. I’ll give you some hints:
- The data set contains 4,906 elements.
- It’s based on a real-world phenomenon from 2015.
- The special points marked by A, B, and C won’t help you identify the data set, but they will be discussed below.
Got a guess?
- The horizontal axis represents “Distance (Feet)” and the vertical axis represents “Frequency.”
Still not sure? Final hints:
- Point A on the graph represents Ruben Tejada’s 231-foot inside-the-park home run on September 2, 2015.
- Point B on the graph represents the shortest distance to the wall in any Major League Baseball park — a mere 302 feet to the right field fence at Boston’s Fenway Park.
- Point C represents the longest distance to a Major League wall — a preposterous 436 feet to the deepest part of center field at Minute Maid Park in Houston.
Okay, you’ve probably guessed by now that the data underlying the graph is the distance of all home runs hit in Major League Baseball during the 2015 season. That’s right, there were 4,906 home runs last year, of which 11 were the inside-the-park variety. The distances ranged from 231 to 484 feet, with the average stretching the tape to 398 feet, and the most common distance being 412 feet (86 HRs traveled that far). The data set includes 105 outliers (based on the 1.5 × IQR convention), which explains why a box plot of the data looks so freaky:
The variety of shapes and sizes of MLB parks helps to explain the data. Like all math folks, I love a good graphic, and this one from Louis J. Spirito at the Thirty81Project.com is both awesome and enlightening:
Here are some more baseball-related trivia you can use to impress your friends at a cocktail party or math department mixer.
- Who holds the record for most inside-the-park home runs in MLB history?
Jesse Burkett, 55 (which is 20 more than he hit outside-the-park)
- Which stadium has the tallest wall?
The left field fence at Fenway Park (a.k.a., the “Green Monster”) is 37 feet tall.
- Which stadium has the shortest wall?
This honor also belongs to Fenway Park, whose right field wall is only 3 feet tall.
- Although only 1 in 446 home runs was an inside-the-park home run in 2015, throughout all of MLB history, inside-the-park home runs have represented 1 in ____ home runs.
- Name all the ways to get on first base without getting a hit.
This is a topic of much debate, and conversations about it have taken me and my friends at the local pub well into the wee hours of the morning. I have variously heard that there are 8, 9, 11, and 23 different ways to get on base without getting a hit. I think there are 8; below is my list.
(2) Hit by Pitch
(4) Fielder’s Choice
(7) Uncaught Third Strike
(8) Pinch Runner
- What is the fewest games a team can win and still make the playoffs?
39. The five teams in a division play 19 games against each of the other four teams in their division. Assume that each of those teams lose all of the 86 games against teams not in their division. Then they could finish with 39, 38, 38, 38, and 37 wins, respectively, and the team with 39 wins would make the playoffs by winning the division.
- Bases loaded in the bottom of the ninth of a scoreless game, and the batter hits a triple. What’s the final score of the game?
1-0. By rule, the game ends when the first player touches home plate.
- In a 9-inning game, the visiting team scores 1 run per inning, and the home team scores 2 runs per inning. What is the final score?
16-9. The home team would not bat in the bottom of the ninth, since they were leading.
Baseball fans do lots of dumb things. Wear a foam finger. Act like they’re smart because they correctly chose the attendance from the four choices on the big board. Throw trash at the lower levels. Streak the field. Root for the Yankees. (And while we’re at it, the 1980’s called. They want “The Wave” back.)
Yet few things are as dumb as the following game, which some friends and I play at Nationals Park.
- To start, everyone puts in a $1 ante.
- The first gambler then adds another $1 to the cup. If the player batting gets a hit, that gambler gets all the money in the cup. If the player batting does not get a hit, that gambler passes the cup to the second person.
- The second gambler then adds $2 to the cup. Again, that gambler gets the money if the player batting gets a hit, or passes the cup to the next gambler if the player batting does not get a hit.
- This continues, with the next gambler adding double what the previous gambler added, until there’s a hit.
Because of exponential growth, it doesn’t take long for this game to exceed most people’s comfort level. At a recent game, we had gone through five batters without a hit when the cup was passed to Dave. He gladly added $32, but then passed the cup to Joe when the batter struck out.
Now that the cup was requesting a three-digit donation, this was getting serious. Most of my friends don’t carry $128 to a baseball game, and even fewer of them are willing to put it in a betting cup. Dave looked at me. “You in?” he asked.
“Sure am,” I replied as I put $100 in the cup. “You’ll have to trust that I’m good for the other $28,” I said. “That’s the last of my cash.”
It occurred to me that 52.9% of the money now in the cup had come from my wallet. That made it extra hard to pass the cup to Adam when the batter struck out.
But Adam just looked at it. “Don’t even think about handing that to me,” he said. “I’m out.”
So the cup went back to Dave, begging for $256 more. “If you’re in, I’m in,” he said to me. “Trust me that I’m good for it?” he asked. I nodded my head to let him know that I did.
When Dave handed the cup to Joe after the next batter flew out, it was Dave’s money that now accounted for 56.3% of the pot. Joe just passed the cup directly to me. “I’m out, too,” he said.
Judgment is the better part of valor. But no one has ever accused me of being valiant. If I put in $512 and there’s a hit, I thought, I’ll win $377. But if not, I’ll be passing a cup to Dave that has $649 of my money.
“I’m in,” I said, surprising even myself. Dave looked at me with equal parts disbelief and dismay… amazed that I had pulled the trigger, unsure what he would do if the cup came to him, stunned that we had gotten to this point.
He’d need to deal with the question sooner than he thought. The batter hit a routine ground ball to the second baseman on the first pitch. As the ball popped in the first baseman’s mitt, my gulp was audible. Folks in our section eyed Dave suspiciously as he cheered an out for the home team. I passed the cup to him and asked, “To you for $1,024, sir. Are you willing to sacrifice your kids’ college fund for this?”
Dave looked at the cup. He looked at me. He looked back at the cup. After several long seconds, he looked at me again. “Split it?” he asked.
I paused. “What?”
“Yeah… what?” said Joe.
Dave explained. “You and I split it. Joe and Adam are out.”
“No!” protested Joe. “That’s not fair! You set us up!”
I ignored Joe, because I realized I now had an exit strategy. “Are you suggesting 50/50?” I asked Dave.
“That’s what I was thinking.”
“But I put two-thirds of the money in that cup.”
“Okay, sure,” Dave said. “We’ll split it 2:1.”
So in the end, I won $33, Dave won $51, and Adam and Joe were disconsolate over what had just transpired.
There had been a streak of 10 batters without a hit (one of the batters had walked, but that doesn’t count as a hit, so our silly game continued), and the pot had grown to $1,027. It would have climbed to $2,051 if Dave had continued. That might only be a drop in the bucket for the Koch brothers, but it’s a sizable amount for a math educator.
The hitless streak made us wonder what game in baseball history would have made us dig deepest into our pockets. The all-time winner would have been a game on May 2, 1917, between the Cincinnati Reds and the Chicago Cubs, when Fred Toney and Hippo Vaughn each threw no-hitters through nine innings. In the top of the 10th, Vaughn got one out before giving up the first hit of the game… so that’s at least 55 batters without a hit. It’s hard to find play-by-play data for games from 1917, but the box score shows that there were 2 walks in the game. If both of them happened before the first hit, that’d increase the pot to a staggering 257 + 3 = $144,115,188,075,855,874. That’s 144 quadrillion, for those of you who, like me, go cross-eyed trying to read such large numbers.
Why do you dislike the number 144?
Because it’s gross.
Why do you dislike the number 144 quadrillion?
Because it’s very gross!
Perhaps most interesting to me, though, was the percent of the cup contributed by the person currently holding the cup. A tabulation of the possibilities is shown below. This scenario assumes four gamblers with no one dropping out. The rows highlighted in yellow (rounds 4n – 3) indicate when the first gambler is holding the cup. Likewise, the second gambler’s turns occur in rows 4n – 2, the third gambler’s turns occur in rows 4n – 1, and the fourth gambler’s turns occur in rows 4n.
|Round||Contribution||Gambler Total||Cup Value||Percent|
Here’s how to read the table:
- The Contribution is the amount that a gambler adds to the cup in that round.
- The Gambler Total is the combined amount contributed by that gambler so far. For instance, the “Gambler Total” in Row 13 is $4,370, because the gambler has contributed 4,096 + 256 + 16 + 1 on his previous four turns, plus the $1 ante at the beginning of the game.
- The Cup Value is the amount in the cup, which includes the ante plus all previous contributions.
- The Percent then shows the percent of the money in the cup that was contributed by the gambler holding the cup.
In Round n, the Cup Value is given by the wonderfully simplistic formula
because the sum of the first n powers of 2 is 2n – 1, and the initial ante contributes $4 more.
More difficult is the formula for the combined contribution shown in the Gambler Total column. For Gambler 1, the formula is:
The Gambler Total formula for gamblers 2, 3, and 4 are similarly complex. (You can have fun figuring those out on your own.) But the purpose of determining those formulas was to investigate the percent of the Cup Value represented by the Gambler Total. And as you can see in the final column, the percent of the cup contributed by the gambler holding the cup tends toward 53.33%.
So here’s the way to think about this. If you pass the cup and the next gambler wins, 53.33% of the money in the cup is his; 26.66% of the money in the cup was yours; and, the remaining 20.00% came from the other two gamblers. Which is to say, you’ve contributed 26.66/46.66 = 57.14% of the winner’s profit.
Taking all this into account, and realizing that I could bankrupt my family if there were ever a streak of just 20 batters without a hit, some additional rules became necessary:
- A person can choose to leave the game at any time. Play continues with those remaining. Obviously, if only one gambler remains, he or she wins.
- Anyone can declare a “cap,” wherein the amount added to the cup continues at the current rate without increasing. For instance, if I added $32 to the cup on my turn and called a “cap,” then every person thereafter would simply need to add $32 on their turn, instead of doubling.
This latter rule seems like a good idea; it transfers the game from exponential growth to linear growth. Still, the pot grows quite rapidly, as shown below:
After 10 consecutive batters without a hit, the pot would still grow to $160, even if a “cap” had been implemented at the $32 mark.
We’ve also switched to the Fibonacci sequence, because that grows less quickly over time.
At a recent Nationals-Cubs game, four of us played this game with the amended rules. Everyone anted $1 to start, and then the amounts added to the cup were:
1, 2, 3, 5, 8, 13, 21 (cap), 21, 21, 21, 21
Dave was third, and the amounts he added to the cup are bold in the sequence above. Both Tanner Roark and Tsuyoshi Wada started strong, and the first 10 batters were retired without a hit or a walk. But the 11th batter of the game got a hit in the bottom of the second inning. Dave had contributed $46 of the $141 in the cup, so he won $95 on that beautiful swing by Wilson Ramos of the Nationals. (It wouldn’t have been “beautiful” if it had been made by a Cub or if it had cost me more than $27.)
The adjustment of using the Fibonacci sequence was a good one. The seventh donation to the cup was a reasonable $21, compared to a $64 contribution using the doubling scheme. In addition, instituting the “cap” permitted folks to continue playing long after they otherwise would have dropped out.
We continued playing through the end of the game. Which brings me to one final rule:
- At the end of the game, the gambler holding the cup does not win. Instead, the money in the cup goes to the next gambler.
This only makes sense. You wouldn’t want a gambler winning if he was holding the cup when an out was recorded.
Last Friday’s game ended with an astonishing 18 hits. There were no other streaks of 10+ batters without a hit, so none of the wins were as large as Dave’s first. On the other hand, I won more than I lost the rest of the game, and I was only down about $10 by game’s end.
Still, I’m concerned that I’ll go broke by the end of the season. One solution, of course, would be to stop playing, but that would require willpower and a higher intellect. Instead, I’m thinking that perhaps I should try to earn some extra money. If you’d like to contribute to this worthy cause, I’m available for tutoring, stand-up comedy, and blog post writing.
And if you have a need for analyzing the dumb games that boys play while watching a baseball game, well, there’s data to suggest that I’m pretty good at that, too.
I’d like to thank Marjan Hong for her help in analyzing this game. I’d also like to thank Dave Barnes for teaching me this game, for wasting countless hours discussing the rules and amendments, and for taking my money.
I was reminded of my second-favorite joke, which is only mildly mathy, while watching the Cubs-Nationals game last night…
What do you do with an elephant who has three balls?
Walk him, and pitch to the rhino.
(If you’re wondering what my favorite joke is, read all about it in Make Your Own (Math) Joke.)
When it comes to pitching and hitting, Jon Lester is clearly better at one than the other. His ERA is an impressive 3.30, but his batting average is .000. That’s right, he’s never gotten a hit in 9 Major League seasons. With two more outs last night — the first, a deep fly ball that was caught by Denard Span, which you can watch on Yahoo sports; the latter, a strikeout — he “improved” to an incredible 0-for-59, a Major League record for futility with the longest hitless streak to start a career.
Lester’s hitless streak is the longest ever by a pitcher. But pitchers aren’t paid to hit. The dubious distinction of the longest hitless streak for a position (non-pitching) player in the Major Leagues is held by Eugenio Vélez, who didn’t get a hit in 46 consecutive at-bats during the 2010-2011 seasons.
A starting player averages 3.3 at‑bats per game, so Velez’s record is equivalent to 14 games without a hit. Assuming that a player is actually trying to hit the ball, a 14-game hitless streak is an impressive accomplishment; and probabilistically, it’s damn near impossible. Not withstanding the likelihood that very few sane general managers would let such a player continue to bat, it also defies the odds that the sun wouldn’t shine at least once on this slumping hitter’s behind.
A while back, I created an Excel file (XLS) to analyze hit streaks. But you could also use it to analyze hitless streaks by changing a couple formulas.
Using Eugenio Velez’s career batting average of .241 (which is deflated, because it includes his record-breaking streak), a hitless streak of 14 games didn’t occur even once in 500,000 games using the Excel sheet model. With 162 games per season, that’s more than 3,000 seasons. Only the very best pro baseball players have a career that spans 20 seasons; those players who hit only .241 have careers that are far shorter, so 46 consecutive at-bats without a hit is impressive, indeed.
Have fun playing with the spreadsheet. Now for a trivia question…
Who is the only Major League player to have 7 hits in one 9-inning game?
Rennie Stennett, Pittsburgh Pirates, September 16, 1975. The Pirates won the game 22‑0 against the Cubs. (Johnny Burnett had 9 hits in an 18-inning game in 1932; three other players have had 7 hits in a game, but all of them required extra innings.)
And a joke…
Why was the calculus teacher bad at baseball?
He was better at fitting curves than hitting them.
And a quote…
Slowest pitch in baseball to reach the catcher? 30 mph, thrown at a 45° angle. Any slower at any other angle hits ground.
— Neil deGrasse Tyson
Pitchers and catchers in Major League Baseball report to spring training this week. Here’s a math joke for baseball players. (And while I’m not certain who said this originally, it’s a solid bet that this quote comes from Dave Barry, though I haven’t been able to confirm that.)
Everyone knows that if a mathematician had to choose between solving a difficult story problem and catching a fly ball, he would surely solve the problem without thinking twice about whether the Infield Fly Rule was in effect.
And since we’re talking baseball…
What do you do with an elephant with three balls?
Walk him, and pitch to the rhino.
Here are some baseball and math trivia questions for you [updated 2/10/2013], some based on problems from Erich Friedman’s Baseball Puzzles page:
- In the bottom of the ninth inning in a game with no score, the bases are loaded. The batter hits a ball that rolls into the right field corner. Before the ball is thrown home, all four base runners cross the plate. What is the final score of the game?
- If the visiting team scores 2 runs per inning, and the home team scores 3 runs per inning, what is the final score of a nine-inning game?
- A player has four at-bats in a game, and he got a hit in his last of these at-bats. His batting averages for the season (rounded to three decimal places, as usual) at three different times during the game do not have any digits in common. What was his batting average at the end of the game?
- What is the minimum number of games a Major League Baseball team must win to make the playoffs?
- The final score is 1-0. Once the first player touches the plate, the home team wins, and the game is officially over.
- The final score is 24-18. With a lead in the middle of the ninth inning, the home team does not bat in the bottom of the ninth.
- At the end of the game, his average was .409. He was 5 for 18 (.278) at the beginning of the game, 6 for 19 (.316) after one at-bat, and 9 for 22 (.409) after three more at-bats.
- A team only has to win 28 games. There are four teams in the American League West division, and the requirement in Major League Baseball is that every team must play either 18 or 19 games against the other teams in their division. That means that each of these four teams could play as few as 54 divisional games. If each of them loses the 108 non-divisional games that they play, then the four teams could conceivably finish with 28, 27, 27, and 26 wins, respectively. Under this scenario, the division winner is the team with just 28 victories.
There are many who complain that, because of the Electoral College, it’s possible for a candidate to be elected President with less than 30% of the popular vote. Yet this situation is worse — a team could make the playoffs by winning just 17% of its games. (Of course, the Presidential election has slightly more consequence. Maybe.)
The day before yesterday, I was at the Nationals-Phillies game. A friend took my sons and me to the game, so in return, I offered to buy dinner at Nationals Park. When I asked his sons what we should get, they answered just as you’d expect any 5- and 9-year-olds to answer: “Pizza!”
I was happy to oblige. I assumed I was getting off cheap.
I was wrong.
At the concession stand, I gasped when I saw that pizza slices were $6 each. Admittedly, they were big slices — an 18″ pizza was divided into 6 slices — but that’s still a lot of money for 42 square inches of pizza. If you think of it as just 14 cents per square inch, it doesn’t feel quite so bad. Until you realize that you could get an entire pie outside the stadium for the cost of 2 slices inside the stadium.
Whatever. The tickets were free, so I ordered a pie. But as I did, I asked the clerk, “Why are pies $36 if slices are $6 each? Shouldn’t there be a discount for buying an entire pizza?” He shrugged his shoulders and gave me the same look I usually get from checkout people when I ask similar questions. His eyes said, “Sorry, dude, I just work here.”
The man behind the counter who was cooking the pizzas must have heard me. When my pie came out of the oven, he used his cutter to divide the pie into 8 slices instead of just 6. I guess he thought I’d feel better if the slices only cost $4.50 each. Never mind that each slice was only 3/4 the size of a regular slice.
Since I felt like I was in the middle of a bad math joke, I figured I ought to deliver the punch line.
“What’d you do that for?” I asked. “I’m not very hungry! There’s no way I’ll be able to eat 8 slices!”
The upside? The Nationals won, 4-2, and they earned the top seed in the playoffs. Go, Nats!
Did you hear about Tommy Lasorda’s wish? The retired manager of the Los Angeles Dodgers is recovering from a heart attack he suffered in June, and he recently told the L. A. Times:
I’ve already told my wife that when I do go, I want our home schedule attached to my tombstone. I want people who are in the cemetery visiting their loved ones to say, “Let’s go to Lasorda’s grave and see if the Dodgers are playing home or away.”
That could get expensive, since it would have to be updated annually. I suggest one of these instead:
Old baseball players never die; they just go batty.
Old baseball players never die; they just do one more lap around the bases.
Old baseball players never die; they just get traded to the Blue Jays. (Sorry, Toronto!)
This got me thinking about what I’d want on my tombstone when I leave.
Don’t meet my end with gasps and shrieks;
I left you with a book (and blog) for geeks.
Given the likelihood of my eternal destination, I take comfort in the following advice:
Go to Heaven for the climate,
to Hell for the company.
Contemplating what should appear on my tombstone puts me in good company. Many mathematicians have pondered the same question.
At age 18, Carl Friedrich Gauss showed that it was possible to draw a regular 17‑gon with compass and straightedge. Proud of his accomplishment, he later requested that a 17‑gon be inscribed on his tombstone. Although his wish was not granted, a memorial to him in his hometown of Braunschweig, Germany, now bears a small 17‑gon just below his right foot.
Imagine a sphere inscribed in a cylinder whose height is equal to its diameter. Archimedes discovered that the ratio of the surface area of the cylinder to the surface area of the sphere is 3:2 and also that the ratio of the volume of the cylinder to the volume of the sphere is 3:2. Despite his many accomplishments in mathematics, this is the one for which he wished to be remembered. He asked that a cylinder with inscribed sphere be displayed on his tombstone, with the ratio 3:2 inside.
Jacob Bernoulli was so enamored with the logarithmic spiral that he wanted one inscribed on his tombstone. Unfortunately, the engraver mistakenly carved an Archimedean spiral (shown below).
An apocryphal story is that the following poem appears on the tombstone of Diophantus. In fact, this problem appeared in a fifth-century Greek anthology of puzzles.
Here lies Diophantus, the wonder behold.
Through art algebraic, the stone tells how old:
“God gave him his boyhood one-sixth of his life,
One-twelfth more as youth while whiskers grew rife;
And then yet one-seventh ere marriage begun;
In five years there came a bouncing new son.
Alas, the dear child of master and sage
After attaining half the measure of his father’s life, chill fate took him.
After consoling his fate by the science of numbers for four years, he ended his life.”
Good luck solving the puzzle before you meet your end!
A well-known problem:
A man walks 1 mile south, 1 mile east, and 1 mile north. He arrives at the same place where he started, and then he sees a bear. What color is the bear?
The answer, of course, is white. It’s a polar bear. These three moves will let a person return to the same place if he starts at the North Pole. (The person could also return to the same place if he starts at an infinite number of points near the South Pole, too. He could start at a point so that when he walks 1 mile south, he is at a point such that the east-west circle on which he is standing has a circumference of 1 mile. Then, he can walk 1 mile east to return to the same spot. Finally, he can walk 1 mile north, and he’s back where he started. Then again, he could also start at a point so that he can walk 1 mile south to a point where the circumference of the east-west circle is 1/2 mile, do that loop twice, then walk 1 mile north. Or find points where the cicumference is 1/3 mile, 1/4 mile, 1/5 mile, etc. You get the idea. However, since there are no bears in Antarctica, the answer to my original question is still correct.)
Two points about this:
- In answer to the question, “Are there polar bears in Antarctica?” there is only one correct answer: Only if they are bipolar.
- I really don’t care to receive silly comments about how a bear trapper could capture a grizzly and take him to Antarctica, or how a brown bear might mistakenly meander north to the Arctic Circle.
Here is a similar question:
A man runs 90 feet, turns left, runs another 90 feet, turns left, runs another 90 feet, and turns left. He is now headed home, and two men with masks are waiting for him. Who are they?
If you don’t know the answer to this riddle, remember that today is the first day of the World Series. My prediction? The Rangers will win easily. It’s not really a fair fight. I mean, members of the Lone Star State’s law enforcement agency with opposable thumbs and automatic weaponry versus defenseless birds? Seriously, if the Rangers don’t win, then we need to seriously reconsider the theory of natural selection.
If you watch the first game of the World Series tonight, remember to enjoy the game. Please don’t get caught up trying to figure out if it converges or diverges.
Here are a few baseball-related math puzzles:
- A baseball player has four at-bats in a game. At three different times during the game, his batting averages for the entire season (rounded to three decimal places) have no digits in common. What was his average at the end of the game?
- During a little league game, the visiting team scored 1 run per inning, and the home team scored 2 runs per inning. What is the final score of this seven-inning game?
- During the first half of the season, Derek batted .100, but his average was .300 during the second half of the season. Similarly, Alex batted .200 the first half of the season and .400 the second half of the season. Both players ended up with the same number of total at-bats, yet Derek had a higher batting average for the entire season. How is this possible?
* Figger Filbert is a term for baseball fans who are obsessed with statistics. Such fans are easily identified; they will make statements like, “Did you know that Albert Pujols is batting .275 when facing married pitchers in suburban ballparks that only sell popcorn on the mezzanine level?” It is a synonym for number nut.