Posts tagged ‘percent’
How awesome was Anthony Davis last night? In a word: very. He set a new All-Star Game scoring record with 52 points, adding 10 rebounds and 2 steals.
(Disclosure: I’ve liked A.D. since he was one-and-done at Kentucky. But it wasn’t until last night that I bought an Anthony Davis jersey:
But as much as I like A.D., I couldn’t help thinking that there needs to be an asterisk next to this new All-Star scoring record. If you only look at points, sure, 52 > 42, so Davis scored more points last night than Wilt Chamberlain scored in the 1962 All-Star Game. But that’s only a part of the mathematical story.
First, let’s talk scoring percentage. In 1962, the final score of the game was 150‑130, meaning that Chamberlain accounted for 15.0% of all scoring. The final score of last night’s game was 192‑182, meaning that Davis accounted for 13.9% of all scoring. Chamberlain gets the nod, but only slightly, and I’ll admit it’s not insignificant that Davis only played 31 minutes last night, while Chamberlain played 37 minutes in 1962. So, maybe this is a push.
But let’s consider shooting percentage. Last night, both teams combined for 55.5% shooting, whereas in 1962, they managed just 43.8% shooting. Perhaps the all-stars from 50 years ago just didn’t shoot as well as players today? Actually, that’s somewhat true: The league FG% for 1961‑62 was 42.6%, the league FG% for 2016‑17 (so far) is 45.6%. But the all-stars last night were 9.9% above the league average, whereas the all-stars in 1962 were just 1.2% above their league average, suggesting that the defense in New Orleans was negligible at best. Which brings me to my next point…
Let’s talk defense. Maybe the combined 374 points that were scored last night doesn’t convince you that defense was nonexistent. Then how about this: In the 1962 game, there were 62 personal fouls. Last night, there were only 16. Even more stark, though: In 1962, all-stars shot 95 free throws during the game; last night, they only shot 8. That’s not a typo, and it’s a pretty clear indication that no one was making much effort to contest shots.
Davis played a great game, but it doesn’t feel right that he unseats Chamberlain, given the circumstances. Not to mention, Chamberlain played a more complete game — shooting 73% from the field, grabbing 24 rebounds, and adding 1 assist.
This brings me to my final point, proportions. The teams last night scored 1/3 more points than their 1962 counterparts, and if you take away that extra third from Davis, he’d have ended the night with 39 points. So if an asterisk is good enough for Maris’s 61 and Flo-Jo’s 10.49, then it ought to be just fine for Davis’s 52, too.
But it is what it is. Congratulations, Anthony Davis.
Looking at the math of basketball is something I get to do quite a bit these days. Discovery Education has formed a partnership with the NBA, and we’re creating a collection of “problems worth solving” using NBA stats and highlight videos. Wanna see some of what we’ve done? Check out www.discoveryeducation.com/NBAMath.
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.
April is Math Awareness Month, and some things to be aware of this month — as well as the whole year through — are common math errors. Here are seven that show up frequently.
Incorrect Addition of Fractions. It’s common for kids to add fractions as follows:
And while that algorithm works for batting averages in baseball, it doesn’t work in most other places. More importantly, this mistake is often unaccompanied by reasoning. For example, a student who claims that 2/3 + 4/5 = 7/9 doesn’t realize that with each addend greater than 1/2, then the sum should be greater than 1. That lack of thought bothers me.
Cancellation of Digits, Not Factors. While it’s true that 16/64 = 1/4 and 19/95 = 1/5, students who think the algorithm involves cancelling digits may also argue that 13/39 = 1/9, and that just ain’t right.
Incorrect Distribution. This one takes a lot of forms. In middle school, kids will say that 4(2 + 3) = 8 + 3. As they get older, they apply the distributive property to exponents and claim that (3 + 4)2 = 32 + 42 or, more generally, that (a + b)2 = a2 + b2.
The Retail Law of Close Numbers. A large portion of the population will buy a shirt for $19.99 that they’d pass up if it had a price tag of $20.00. Even though the amounts only differ by one cent, a lesser digit in the tens place makes the price feel much lower. Crazy, but true.
Ignoring the Big Picture. If you are a driver who is interested primarily in speed (and less concerned with price, looks, fuel efficiency, or other factors), would you rather have a vehicle with 305 horsepower or one with 470 horsepower? If you chose the latter option, congratulations! While the owner of a sweet 305-hp Ford Mustang will be sitting at home and sipping a mint julep on his front porch, you’ll still be doing 30 mph on the highway in your Sherman tank.
Correlation Implies Causation. As ice cream sales increase, the number of drowning deaths increases, too. But that doesn’t mean that having an ice cream cone willl make you less likely to swim safely, even if you failed to heed your mother’s advice to wait 30 minutes after eating. It’s just that ice cream sales and swimming-related deaths increase in summer, both of which are to be expected.
Just because two things happen to coincide doesn’t mean that one is the direct (or even indirect) result of the other.
Percents Don’t Work That Way. A 20% decrease followed by a 20% increase does not return you to the initial value. If you invest $100 in a company, and it loses 20% the first year, your investment will then be worth $80. If it gains 20% the next year, you’ll now have $96. Uh-oh.
What common math error do you see frequently, and which one bothers you the most?
So, I understand what Baltimore Ravens coach John Harbaugh meant when he said:
I never thought you could feel 100% elation and 100% devastation at the same time. But I learned tonight you can.
But it sure sounds to me like he has twice as much capacity for emotion as the rest of us. It reminds me of Yogi Berra’s famous quote:
Baseball is 90% mental, and the other half is physical.
Or the anonymous quote about our favorite subject:
Mathematics is 50% formulas, 50% proofs, and 50% imagination.
Dare to guess what percent of Americans wouldn’t be able to identify the math errors in those statements?
Here’s a good old-fashioned math joke involving percents:
What’s a proof?
One-half percent of alcohol.
And a slightly longer one:
“Statistics is wonderful!” said a statistician.
“How so?” asked his friend.
“Well, according to statistics, there are 42 million alligator eggs laid every year. Of those, only about 50% hatch. Of those that hatch, 75% are eaten by predators in the first 36 days. And of the rest, only 5% get to be one year old for one reason or another.”
“What’s so wonderful about that?”
“If it weren’t for statistics, we’d be up to our asses in alligators!”
I stumbled across the maps of the problematic blog last week, which claimed the U.S. Senate was no longer necessary. It was part of a list of eleven unnecessary things, actually. The list also included phone books, beepers, the Electoral College, and pocket calculators. The author claimed that the Senate gives just 18% of the U.S. population the power to stop a bill from passing Congress. That is, if 50 Senators vote “no” to a bill, then it fails, and the 25 least populous states represent just 18% of the population. I didn’t check the author’s math, but I’ve heard similar estimates before, so 18% sounds reasonable to me.
Interestingly, the author implied that the Senate might have been necessary when it was first created, to give a voice to smaller states. This made me wonder — when Congress first began enacting law in 1789, what percent of the U.S. population had the power to stop a bill?
The first Congress had only 24 Senators from 12 states. (Rhode Island originally rejected the Constitution in 1788, delaying ratification until May 1790, when the federal government threatened to treat them as a foreign government.) Consequently, the Senators from six states had the power to stop a bill.
The populations (in thousands) of the 12 states with Senators in 1789:
- Conecticut: 237
- Delaware: 59
- Georgia: 82
- Maryland: 96
- Massachusetts: 379
- New Hampshire: 142
- New Jersey: 184
- New York: 340
- North Carolina: 393
- Pennsylvania: 434
- South Carolina: 249
- Virginia: 747
The total population (in thousands) of those 12 states was 3,342. The total population of the six least populous states was 568. That means that 568/3,342 ≈ 17% of the U.S. population could have stopped a bill in 1789.
Please understand, I’m not arguing that the Senate should be retained or abolished. But by the numbers, it appears that the Senate might have been even less necessary in 1789 than it is today.
Anyway, here’s a joke about math and politics:
A cannibal goes to the butcher shop and notices that mathematician brain is selling for $1 a pound, but politician brain is selling for $4 a pound. “Is the politician brain really that much better?” she asks the butcher.
“Not really,” he says. “But it takes a whole lot more politicians to make a pound.”