How Not To Be Wrong
; Jordan Ellenberg
This is just a book review.
These are not my words.
> If you play soccer, you’ve got to do a lot of boring, repetitive, apparently pointless drills. Like this, math is the same. With the tools of mathematics in hand, you can understand the world in a deeper, sounder, and more meaningful way.
[Abraham Wald Story During Wolrd War II]
At the applied mathematics gropu Columbia, dozens of young women bent tover merchant desktop calculators were calculating formulas for the optimal curve a fighter should trace out through the air in order to keep an enemy plane in its gunsights. In another apartment, a team of researchers from Princeton was developing protocols for strategic bombing.
You don’t want your planes to get shot down by enemy fighters, so you armor them. But armor makes the plane heavier, and heavier planes are less maneuverable and use more fuel. Somewhere in between there is an optimum.
When American planes came back from engagements over europe, they were covered in bullet holes. The officers saw an opprotunity for efficiency; you can get the same protection with less armor if you concentrate the armor on the plate with the greatest need, where the planes are getting hit the most. But exactly how much more armor belonged on those parts of the plane? That was the anwer they came to Wald for. However, it wasn’t the answer they got. Wald said that the armor doesn’t go where the bullet holes are. He said, “It goes where the bullet holes aren’t, which is on the engines.” Wald knew that the missing bullet holes were on the missing planes. The reason planes were coming back with fewer hits to the engine is that planes that got hit in the engine weren’t coming back.
> Countries don’t win wars just by being braver than the other side, or freer, or slightly preferred by God. The winners are usually the guys who get 5% fewer of their planes shot down, or use 5% less fuel, or get 5% more nutirition into their infantry at 95% of the cost.
> Why did Wald see that the officers, who had vastly more knowledge and understanding of aerial combat, couldn’t? It came back to his math-trained habits of thought.
> Mutual funds don’t live forever; some flourish, some die. Judging a decade’s worth of mutual funds by the ones that still exist at the end of the ten years is like judging our pilots’ evasive maneuvers by counting the bullet holes in the planes that came back.
> Using limit formula, Wald calculated the amount of armor plates for planes during World War II.
> How can we prove A + B = B + A ? We can make the first decision and the second decision. Let the first decision A and the second decision B. You perform the task A and B. Or you perform the task B and A. That makes A + B = B + A.
> What about A * B = B * A? No matter which ways you look at a rectangle, it is still the same rectangle.
> Jordan Ellenberg comments that algebra is the most practical math filed for normal people.
> “Why is Obama trying to make America more like Sweaden when Swdes are tyring to be less like Sweden?”
Let’s use y = -x + c
c: the maximum prosperity
USA (x = 3)
Swden (x = 7)
According to the chart, the more Swedish you are, the worse off your country is. The Swdes have figured this out and are launching their northwest climb towrd free market prosperity. However, Obama is sliding in the wrong direction.
Let’s use another graph.
y = prosperity
x = swedishness
b = the maximum prosperity
y = – (x – a)(x – a) + aa
USA x and y values are lower than Sweden’s.
Swden’s x and y values are a little ahead of the peak point.
The optimum is somewhere in between.
y = revenue
x = tax rate
y = – (x – 50)(x – 50) + 2500
The medium is the best.
The tax rate is 0%; in that case, the government gets no tax revenue.
The tax rate is 100%; whatever income you have, that goes straight into uncle Sam’s bag, which is nothing.
In the intermediate range in the middle of the curve, where the government charges us somewhere between none of our income and all of it – in other words, in the real world the government does take in some amount of revenue. That means the curve recording the relationship between tax rate and government revenue can not be a straight line. The Laffer Curve is 90% correct. Gret Mankiw restates that the graph is not necessarily nonlinear, but if it slopes upoward in one place, it has to slope downward somewhere else.
> The area of a circle is (pi)rr. Let r = 1, then the area is (pi).
> The Pythagoreans believed that odd numbers were good and even numbers evil.
> Let’s draw the inscribed square. The square is in a circle. Jordan Ellenberg proves the Pythagorean Teorem.
A. y = -x + 1
B. y = x – 1
C. y = -x – 1
D. y = x + 1
1) yy = (-x + 1)(x – 1)
= – (x – 1)(x – 1) proven
2) yy = (-x – 1)(x + 1)
= – (x + 1)(x + 1) proven
> Jordan Ellenberg also proves the area of a triangle. One square’s half is a triangle, which is ½. y = x separates the square in half, so the ingegration is y = 1/2(xx). Proven
> Pi is a complete circle. Pi is between 2 and 4. The circumscribed square is 4. The inscribed octagon is 2.83. The circumscribed octagon is 3.31. Therefore, the area of the circle is between 2.83 and 3.31.
> If you keep zooming in a curve, it becomes a straight line.
> The circle actually is a polygon with infinitely many infinitely short sides.
> There is Zeno’s theory. Zeno keeps walking to the destination. He keeps walking tiny bits after tiny bits. However, the distance is still far. He concludes now that it is impossible to approach to the destination.
This will be something like this: ½ + ¼ + 1/8 + 1/6 + 1/32 + … It is more like 0.99999… You never go to 1.
However, 0.9999… is actually one. 0.33333 is 1/3 and we can multiply three, so 1/3 becomes 1, which is the same as 0.99999….
> (10 * 0.9999…) – 0.999999…
→ 9.9999… – 0.9999…
This means that 9 * 0.9999… = 9
> Let’s look at the other one.
1 + 2 + 4 + 8 + 16 + … = infinite
2 * (1 + 2 + 4 + 8 + 16 + …) is (2 + 4 + 16…). This one is one less than (1 + 2 + 4 + 8 + 16 + …). So, 2 * (1 + 2 + 4 + 8 + 16 + …) – (1 + 2 + 4 + 8 + 16 + …) = -1
Then, with simplication we get 1 + 2 + 4 + 8 + 16 + … = -1. However, I Alex Kim looks at this different. When you multiply 2 to the original one, you get double more numbers, so the answer is infinite.
> Let’s see 1 – 1 + 1 – 1….
→ (1 – 1) + (1 – 1) = (1 – 1)… = 0 + 0 + 0 + …
As well, 1 – 1 + 1 is equal to 1 – (1 – 1).
So, 1 – 1 + 1 – 1 + 1 – 1 … becomes 1 – (1 – 1) – (1 – 1) – (1 – 1)…
= 1 – 0 – 0 – 0…
I Alex Kim corrects this like that: (1 – 1 + 1) – 1 + (1 – 1 + 1)… = 0.
> Let’s see another series of Jordan Ellenberg.
T = 1 – 1 + 1 – 1 + 1…
-T = – 1 + 1 – 1 + 1 – 1…
T – 1 = -1 + (1 – 1 + 1 – 1 + 1…
So, -T = T – 1
This is what Jordan Ellenberg claims.
I Alex Kim looks at this differently.
T = 1 – 1 + 1 – 1 + 1…
-T = – (1 – 1 + 1 – 1…
T – 1 = (1 – 1 + 1 – 1 …) – 1
So, I Alex Kim believe that T = 0 and -T = 0 as well. Then, T – 1 is -1, which makes a sense.
> Let’s discuss that 0.99999… is one or not.
.9 + .09 + .009 + .0009 + …
Jordan Ellenberg claims that this series goes to one eventually.
> Jordan Ellenberg claims that by 2048 all Americans will become overweight. However, he also claims that we are not going to be overweight in the year 2048.
> Let’s go back to the T equation.
T = 1 – 1 + 1 – 1 + …
-T = -1 + 1 – 1 + 1…
T – 1 = -1 + (1 – 1 + 1 – 1…)
So, -T = T – 1 is wrong.
1. a = b
2. aa = ab
3. aa – bb = ab – bb
4. (a – b)(a + b) = b(a – b)
5. a + b = b
6. 2b = b
Number four is wrong. a – b = 0. This is very similar to Jordan Ellenberg’s T equations. Remember that -T = 0 Simply, -T = (-1 + 1) – 1 + 1 – 1 + 1…
> Every extra $10,000 of income makes you 3% more likely to vote Republican.
> You might think schools with higher SATs are liekly to be pricier; but a look at the data tell you that’s not a universal law.
y = tuition
x = average SAT score
y = 28x + $10,000
> Sometimes you need to go without calculators.
> We have to teach a mathematics that values precise answers but also intelligent approximation.
[Jordan Ellenberg’s American Obesity Discussion]
Here is the American obesity rate: 1970 (50%), 1990 (60%), and 2008 (75%).
y = obesity rate
x = year
If this graph is drawn linearly, by 2048 everybody will be obesed. By 2060, 109% of Americans would be overweight. In reality, the graph is a parabola. Jordan Ellenberg also claims that Black men are less likely to be overweight.
> Let’s discuss the death rates.
y = a number of killed
x= a population (a million)
y = 150x for slovenia (2,300)
y = 537/3 (x) for Israel (6,1074)
> California, Texas, New York, and Florida have the most brain cancer because they have the most people. This is why we need to be careful with percentages or rates. Instead of counting raw numbers of brain cancer deaths by state, we can comp¨te the proportion of each state’s population that dies of brain cancer each year. South Dakota is the first place. Then, Nebraska, Alaslca, Delaware, and Maine. These are the places to aviod if you don’t want to get brain cancer. So, you might move to Wyoming, Vermont, North Dakota, Hawaii, and the District of Columbia. However, South Dakota sin’t necessarily casuing brain cancer, and North Dakota isn’t necessarily preventing it. If the population is high less likely the results come and if the population is low more likely the results come. Essentially, the rate of death up and down around 50% each state. The more coins you flip, the more and more extravagantly unlikely it is that you will get 80% heads. Measuring the absolute number of brain cancer deaths is biased toward the big states; but measuring the biggest rates of the lowest ones.
> Let’s have a basketball discussion. For 2011 to 2012, five players were locked in a tie for the highest shooting percentage in the league: Armon Johnson, DeAndre Liggins, Ryan Reid, Hasheem Thabeet, and Ronny Turiaf. These were not the five best shooters in the NBA. These were people who barely ever played. Armon Johnson appeared in one game and took one shot. You wouldn’t claim that Armon Johnson was a more accurate shooter than the highest-ranking full-time player on the list, Tyson Chandler, who made 141 out of 202 shots over the same time period.
> Let’s discuss schools. This time a small school did not dominate the twenty-five because that school hand more variable test scores. In a large school, the effect of a few extreme scores will simply dissolve into the big average, hardly but getting the overall number. The first ten flips become less and less important the more flips we make. It is not by balancing out what’s already happened, but by diluting what’s already happened with new data utnil the past is so proportionally negligible that it can safely be forgotten.
> The question of whether one war was worese than another is fundamentally unlikely the question of whether one number is bigger than another.
> Proportions can be misleading even in simpler, seemingly less ambiguous cases.
> Negative numbers are numbers but treated differently. Because of negative numbers, percentages can be often incorrect. For that reason, you can’t trust politicians. The statstics does not make a sense.
Here is an example: Ultrarich 37%, Rich 56%, Normal 99%.
Negative numbers in the mix make percentages act wonky. You can’t put 131% in a pie chart.
> “The Obama administration has brought hard times to American woemn.” It is true with an official record. However, those numbers are net job loses. We have no idea how many jobs were created and how many destroyed over the three-year period. The net job loss is positive sometimes, and negative other times, which is why taking percentages of it is a dangerous business. Actually, men lost mucy more jobs than women. For that reason, it is hard to trust public statistics. Jordan Ellenberg says that the media fools us.
> A real-world problem is somethin glike “Has the recession and its after math been especially bad for women in the work force, and if so, to what exten is this the result of Obama administration policies?” For this, you need to know more than numbers.
1. What shape do the job-loss curves for men and women have in a typiclas recession?
2. Was this recession notably different in that respect?
3. What kind of jobs are disproportionatelyh held by women, and what decisions has Obama made that affect that sector of the economy?
> Now you skip forwards 613 letters (why 613? because that’s the exact number of commandments in the Torah) and start counting every fifth letter again. You find that letters spell out Torah.
> Robert E. Kass says that Bible is a challenging puzzle.
> The American journalist Michael Drosnin went hunting for LES, of his own, jettisoning scientific restraint and counting every cluster of sequences he could find as a divine foretelling of future events. In 1997, he published ‘The Bible Code.’
> Here is a parable. One day, you receive an unsolicited newsletter from a stockbroker in Baltimore, containing a tip that a certain stock is due for a big rise. A week passes and just as the Baltimore stockbroker predicted, the stock goes up. The next week, you get a new edition of the newsletter, and this time, the tip is about a stock whose price the broker thinks is going to fall. And indeed, the stock craters. Then weeks go by, each one brining a new issue of the mysterious newsletter with a new prediction, and each time, the prediction comes true. On the eleventh week, you get a solicitation to invest money with the Baltimore stockborker, naturally with a hefty commision to cover the keen veiw of the market so amply demonstrated by the newsletter’s ten week run of golden picks. Jordan Ellenberg expalins this with mathematics. The duffer has a 50% chance to get. Later, ½ * ½ * ½ * ½ * ½ * ½ * ½ * ½ * ½ * ½ = (1/1024)
The cahnce that a duffer would do so well is next to nil. Here is what you didn’t see the first time.
That first wee, you weren’t the only person who got the broker’s newsletter; he sent out 10,240. But the newsletters weren’t all the same. Half of them were like yours, predicting a rise in the stock. The others predicted exactly the opposite. The 5120 people who got a dud prediction from the stockbroker never heard from him again. But you, and the 5119 others who got your version of the newsletter, get another tip next week. Of those 5120 newsletters, half say what yours said and half say the opposite. And after that week, there are still 2,560 people who’ve received two correct predictions in a row. And so on. After the tenth week, there are going to be then lucky people who’ve gotten ten straight winning picks from the Baltimore stockbroker.
> Jordan Ellenberg Suggests that they use strategies when a company launches a mutual fund. He suggests that it is very dangerous to invest money in mutual funds. Brokers and advisers paly games on you.
> It’s massively improbable to get hit by a lighning bolt, or to win the lotter; but these things happen to people all the time.
> A dead fish, scanned in an fMRI device, was shown a series of photographs of human beings, and was found to have a surprisingly strong ability to correctly assess the emotions the people in the pictures displayed.
> Let’s see a missile equation.
Height = 100 + 200x
100 = the missile launched from 100m above
200 = upward velocity of 200m/s
With gravity, pushing down, height = 100 + 200x – 5xx
For when it lands, 100 + 200x – 5xx = 0
With the quadratic formula, you can solve this.
The negative solution is -.4939015319… means that the missile is launched probably eariler.
> Cubic formula doesn’t exist and is disproven.
> Probablity does not bring answers. For example, what is the probablity that counsuming olive oil prevents cancer? Another example is that what is the probablity that Shakespeare was the author and Shakespeare’s plays?
> The names of medieval rabbis are hidden in the letters of the Torah.
> It’s not enough that the data be consistent with your theory; they have to be inconsistent with the getation of your theory.
1. Run an experiment.
2. Suppose the null hypothesis is true, and let p be the probablity (under that hypothesis) of getting results as extreme as those observed.
3. The number P is called the p-value. If it is very small, rejocie; you get to say your results are statistically significant. If it is large, concede that the null hypothesis has not been ruled out.
> A mathematician John Arbuthnot studied the records of children born in London between 1629 and 1710, and found there a remarkable regularity: in every single on of these eight-two years, more boys were born than girls. Then the probality in any given year that London would welcome more boys than grils would be ½; and the p-value, the probablity of the boys winning eighty-two times in a row is ½ * ½ * ½ * … 82 times … * ½
in other words, more ore less zero. Nicholas Bernoulli proposed a different null hypothesis: that the sex of a child is determined by chance, with an 18/35 chance of being a boy and 17/35 of being a girl.
> Mathematics has a funny relationship with the English language.
> When we are testing the effect of a new drug, the null hypothesis is that there is no effect at all; so to reject the null hypothesis is merely to make a judgement that the effect of the drug is not zero. But the effect could still be very samll-so small that the drug isn’t effective in any sense that an ordinary non-mathematical Anglophone would call significant.
> Twice a tiny number is a tiny number.
> You can’t trust studies without the null hypothesis.
> Shakespeare is famouse as a master of the alliterative line, in which several words in close succession start with the same sound (“Full fathom five thy father lies…”)
> Past performance is no guarantee of future returns.
> Shot locations are more important than shot successes. This explains more clear studies.
> What we are trying to prove, in most case, is that the null hypothesis isn’t true.
1. Suppose the hypothesis H is true.
2. It follows from H that a certain fact F can not be the case.
3. But F is the case.
4. Therefore, H is false.
[How can we prove that the square root of 2 is an irrational number?]
1. The squre root of 2 = m/n
2. 2 = mm/nn
3. 2nn = mm
→ mm is an even number.
Also, m itself is an even number.
For that reason, the square root of 2 is not a rational number.
[The Null Hypothesis Significance Test]
1. Suppose the null hypothesis H is true.
2. It follows from H that a certain outcome O is very improbable.
3. But O was actually observed.
4. Therefore, H is very improbable.
> Impossible and improbable are not the same. Impossible things never happen. But improbable things happen a lot.
> The appearance of start clusters in random data offers insight even in situatoins where there is no real randomness at all, like the behaviour of prime numbers. In particular, Prime numbers get less and less common as the numbers get bigger. The gaps between the even numbers are always exactly of size 2. For the powers of 2, it’s a different story. But the question of gaps between consecutive primes is harder.
> N / log N is a prime number.
> Although the numbers are getting bigger, there are lots of twin primes. For exmaple, 11 and 13 are twin primes because the gap between them are two.
[Jordan Ellenberg proves how certain gens influence schizophrenia.]
1. Out of the large population of genes, 94990 do not pass the p-value test for no effect.
2. Out of the medium population of genes, 5000 pass the p-value test for no effect.
3. Out of the tiny population of genes, 5 do not pass the p-value test for no effect.
4. Out of the tiny population of genes, 5 pas the p-value test for the effect.
→ The genes that don’t affect schizophrenia very rarely pass the test, while the genes we are really interested in pass half the time.
> A lot of time researches are wrong.
> Genomicists nowdays believe that heritability of IQ scores in probably not concentrated in a few smarty-pants genes, but rather accumulates from numberous genetic features, each one having a tiny effect. This means that if ou go hunting for large effects of individual polymorpnisms, you will succeed – at the same 1-in-20 rate as do the entrail readers.
[Jelly Beans Cause Acne]
1. Jelly Beans cause Acne.
2. Scientists investigate.
3. We found no link between Jelly Beans and Acne.
4. Probably certain colors?
5. We found no link between purple jelly beans and acne.
6. We found no link between brown jelly beans and acne.
7. We found no link between pink jelly peans and acne.
8. We found a link between green jelly beans and acne.
9. News: Green Jelly Beans linked to Acne!!!
But what if the green jelly beans were tested twenty times by twenty different research groups in twenty different labs?
> A result with a weak p-value may provide only a little evidence, but a little is better than none; a result with a strong p-value might provide more evidence, but as we’ve seen, it’s for from a certification that the claimed effect is real.
> When a durg fails a signifificance test, we don’t say, “We are quite certain that the drug didn’t work,” but merely, “The drug wasn’t shown to work.”
> The significance test is the detective, not the judge.
> The more measurements you can make and the more precise those measurements are, the better you’re going to do at pinning down its track. However, some problems are more like predicting the weather. Human behaviour ought to be even harder to predict than weather. For human action we have no such model and may never have one.
[Does Facebook know you’re a terrorist?]
> Facebook generally knows its user’s real names and locations, so it can use public records to generate a list of Facebook profiles belonging to people who have already been convicted of terroristic crimes or support of terrorist groups.
> There are 200 million American Facebook users.
> 100,000 Facebook usuers are involved in terrorisms and 10 people are terrorists.
> 199,990,000 are nonoffenders in Facebook and they are not on the list.
> 9,990 are terrorists and they are not on the list.
→ 99,990 / 199,990,000 = .05%
There are two questions.
1. What’s the chance that a person gets put on Facebook’s list, given that they’re not a terrorist?
2. What’s the chance that a person’s not a terrorist, given that they are on Facebooks’s list?
> When you’re trying to decide whether your neighbor is a secret terrorist, you have critical prior information, which is that most people aren’t terrorists.
> Nobody guessed RRRR, although it showed up a lot. A lot of people even guessed BBRBR.
> If you ask people to pick a number between 1 and 20, 17 is the most common choice. If you ask people for a number between 0 and 9, they most frequently pick 7.
> People are hard to be random.
> When we vote someone for elections, we tend to pick a good number than a person.
> There is a wheeling game dropping five balls. If certain colors of balls drop more regularly, the probablity goes more weird and changes.
> Here is a story of Jordan Ellenberg. Everyone around thought my friend was the dirtiest man in school, because he wore the same T-shirt every single day. But in fact, he was the cleanest man in school, dressed every day in a new-from-the-store, never-worn shirt!
> Generally, buying more lottery tickets make you a less loser.
→ 9,999,999 / 10,000,000 times: ticket is worthy nothing
→ 1 / 10,000,000 times: ticket is worthy $6 million
What is the value of a lottery ticket?
9,999,999 / 10,000,000 * $0 = $0
1 / 10,000,000 * $6,000,000 = $0.60
$0 + $0.60 = $0.60
So, it’s 60 cents. However, we certainly don’t expect the lottery ticket to be worthy 60 cents. The Powerball lottery ticket costs $2, but the actual value is 94 cents.
> What is the chance that all 75 million of your competitors lose (Jackspot)?
(174,999,999 / 175,000,000) power 75 million = 0.651
So, it will be a 65% chance that none of your fellow players will win, which means there is a 35% chance at least one of them will win. Jordan Ellenberg advises to avoid playing Powerball.
> When you buy a Cash WinFall ticket on a roll-down day, your ticket is most likely a loser. But on roll-down day, the prizes, in the unlikely event that you do win, are bigger.
> X and Y are to numbers whose values we’re uncertain about.
E(X) is the expected value of X.
So, E(X + Y) = E(X) + E(Y)
There is a game called franc-carreau, which is throwing the coin on the floor and make a bet: will it land wholly within one tile, or end up touching one of the cracks? Buffon discusees this game.
The coin has radius r. The square tile has a side of length L. The coin touches a crack exactly when its center lands otusde a smaller square, whose side has length L-2r. The smallest square has area (L-2r)(L-2r), while the bigger square has area LL. The winning chance is (L-2r)(L-2r) / LL
→ ½, which is a 50% chance
[Buffon’s Needle Problem]
Suppose you have a hardwood floor made of long, skinny slats, and you happen to have in your possession a needle exactly as long as the slats are wide. Throw the needle on the floor. What is the chance that the needle crosses one of the cracks separating the slats? We need to keep track of not just where the center of the needle falls, but also what direction it’s pointing. The probablity is 2/pi or about 64%.
Probablity P that the thrown-down needle crosses a rack.
(1-P) that the needle doesn’t cross any cracks.
(1-P) * 0 = 0
P * 1 = P
The expected number of crossing is simply P.
When you’re faced with a math problem you don’t know how to do, you’ve got two baisc options. You can make the problem easier, or you can make it harder.
What if we ask, more generally, about the expected number of crack crossing by a needle that is two slats wide? There are three possiblities.
1. The needle land entirely within one slat.
2. It could cross one crack.
3. It could corss two.
→ This become 2P.
> Do not throw away your lottery tickets.
> You will be always possible to miss your plane.
> We need to eliminate every wasting.
> Pascal believed in God.
>Let’s play the head and tail game. You have 50% to get the tail or the head. So, (½)*1 + (1/4)*2 + (1/8)*4 + (1/16)*8 …
This becomes ½ + ½ + ½ + …
The game never ends.
> Money and utility graph goes with logarithm.
> Jordan Ellenberg shows another math trick.
½ + ¼ + 1/8 + 1/16 + 1/32 + … = 1
¼ + 1/8 + 1/16 + 1/32 + … = ½
1/8 + 1/16 + 1/32 + … = ¼
1/16 + 1/32 + … = 1/8
1/32 + … = 1/16
And ½ + 2/4 + 3/8 + 4/16 + 5/32 + … = 2 when the all series above are added together.
> Suppose there is an urn with 90 balls inside. You know that 30 balls are red; concerning the other 60 balls, you know only that some are black and some are yellow.
RED: You get $100 if the next ball pulled from the run is red; otherwise, you get nothing.
BLACK: You get $100 if the next ball is black, otherwise nothing.
NOT-RED: You get $100 if the next ball is either black or yellow, otherwise nothing.
NOT-BLACK: You get $100 if the next ball is either red or yellow, otherwise nothing.
With RED, you know where you stand. You’ve got a 1-in-3 chance of getting the money.
With BLACK, you have no idea what odds to expect. As for NOT-RED and NOT-BLACK, the situation is just the same. Mr. Ellenberg’s subjects liked NOT-RED better, preferring the state of knowing that their chance of a playoff is exactly 2/3. The “knwon unknowns” are like RED – we don’t know which ball we’ll get, but we can quantify the probablity that the ball will be the player to an “unknown unknown” – not only are we not sure whether the ball will be black, we don’t have any knowledge of how likely it is to be black.
> Would you rather have $50,000 or would rather have a 50/50 bet between losing $100,000 and gaining $200,000? The expected value of the bet is (½) * (-$100,000) + (½) * ($200,000) = $50,000
> In Euclid’s plane, two different points determine a single line, and two different lines determine a single intersection point – unless they’re parallel. Lottery is like a geometric figure.
> This is Hamming’s definition: the distance between two blocks is the number of bits you need to alter in order to change on block into the other. For example, the distance between 0010111 and 0101011 is 4.
> The set of points at distance less than or equal to 1 from a given central point is a circle and a sphere.
> Instaed of choosing a combination with your own, you can calculate the whole list of combinations and choose it. In order to play the lottery without risk, it’s not enough to play hundreds of thousands tickets; you have to play the right hundreds of thousands of tickets. “I [Jordan Ellenber] was able to get in touch with Yuran Lu, but he didn’t know exactly how those tickets had been chosen; he told me only that they had a “go-to guy” in the dorm who handled al such algorithmic matters. I [Jordan Ellenberg] can’t be sure whether the go-to guy used the Denniston sytem or something like it. But if he diodn’t, I think he probably should have.
> Secrist concluded that clothing stores are on average financially.
> Galton claims that they are not parents who determine the heights of the offspring. He also claims that gifts are same things. Height, Galton understood, was determined by some combination of inborn characteristics and external forces.
> The companie that were lucky in 1922 were no more liekly than other companies to be lucky ten years later.
> Matt Kemp is the superstar hitting. .460 and on pace for 86 home runs, 210 RBIs, and 172 runs socored. Emp hit nine home runs in the Dodgers’ first seventeen games.
H = G * (9/17)
H = the number of home runs kemp hit for the full season
G = the number of games his team plays
A baseball season is 162 games long. So, 85.7647 = 162 (9/17)
However, no one at ESPN thinks Matt Kemp is going to hit 86 home runs.
> I [Jordan Ellenberg] looked at first-half American League home runs leaders in nineteen seasons between 1976 and 2000. Even there is a regresswion going on in baseball games. A lot of those players went through regression. Even when Kemp came back from the injury, he became the different person.
> We have slow days and fast days. This is proven by experiements. Child behaviours can be same things.
> Most of the British soldiers lost in the Crimean War had been killed by infectons, not Russians.
> y = x when every son and father have equal heights. In reality, y = x becomes an ellipse diagonally.
y = son’s height
x = father’s height
The sons are on average shorter than father. When the son’s height is completely unrelated to those of the parents, Galton’s ellipses are all circle, and the scatter plot looks roughtly around. When the son’s height is completely determined by heredity, the data lies along a straight line.
> Kerry vs Obama
y = Obma 2008 vote share
x = Kerry 2004 vote share
y = x but dots are a little higher. So, Obama was doing better.
> GE daily price change and Google daily price change are almost zero.
> Mostly low-income people voted for George W. Bush in 2004.
> Axx + Bxy + Cyy + Dx + Ey + F = 0 determines three main classes.
xx + xy + yy – 1 = 0 is esslipse
xy – 1 = 0 is hyperbola
yy – x = parabola
> If regression to the mean is a universal phenomenon, why don’t temperatures do it too? In fact, it does according to weather data.
> The correlation between the two variables is determined by the angle between the two vectors. And it is cosine of the angle.
1. (married smokers / all married people) < (all smokers / all people)
2. (married somkers / all smokers) < (all married people / all people)
→ This becomes (married smokers * all people) < (all smokers * all married people)
Marriage is connected with smoking. Yes, negatively.
>There is a mystery factor.
> Lung cancer has been increasing. Don and Hill’s study shows that lung cancer causes smoking.
> Fisher claims that smoking appears to be subject to at least some heriatable effects.
> By 1964, the health hazard of smoking is revealed.
> Berkson was a vigorous skeptic about the link between tabacco and cancer.
> Jordan Ellenberg uses a mathematical approach. He claims that the correlation between smoking and lung cancer is same as the correlation between high blood pressure and diabetes. According to his research, high blood and diabetes are correlated but also surprisingly not related. It is more likely to have both.
> There are three categories: raise taxes, cut defense, and cut medicare. We want to cut, but we also want each program to keep all its funding. The average American thinks there are plenty of non-worthwhile federal programs that are wasting our money. There are three chocies: leave the health care law alone, kill it. Or make it stronger. And each of the three chocies is opposed by most Americans.
> Slime molds make decisions.
> There is a list of all three candiates in the preferred order.
Bush, Gore, Nader 49%
Gore, Nader, Bush 25%
Gore, Bush, Nader 24%
Nader, Gore, Bush 2%
The first group represents Republicans and the second group liberal Democrats. The third gropu is conservative Democrats for whom Nader is a little too much. The fourth group is people who ovted for Nader. You can give each candidate points according to their placement: if there are three candidates, give 2 for a first-place vote, 1 for second, 0 for third. In this scenario, Bush gets 2 points from 49% of the voters and 1 point from 24% more. So, 2 * 0.49 + 0.24 = 1.22 Gore gets 2 points from 49% of the voters and 1 point from another 51%. The score is 1.49. And Nader gets 2 points from the 2% and another point from the liberal 25%. The score is 0.29. SO, the raink will be with Gore, Bush, and Nader. So 51% of the voters prefer Gore to Bush, 98% prefer Gore to Nader, and 73% prefer Bush to Nader. But what if the numbers were slightly shifted?
Say you move 2% of the voters from “Gore, Nader, Bush” to “Bush, Gore, Nader.”
Bush, Gore, Nader 51%
Gore, Nader, Bush 23%
Gore, Bush, Nader 24%
Nader, Gore, Bush 2%
Now, a majority of Loridians like Bush better than Gore. The slime mold mentioned above, represents human natures.
> Florida 2000 would have looked like under the Australian system.
> There is a theory of Condorcet. “If the majority of voters prefer candidate A to candidate B, then candidate B can not be the people’s choice.
> If P is a point and L is a line not passing through P, there is exactly one line through P parallel to L. (We will see.)
> There is a line joining any two points.
> Any line segment can be extended to a line segment of any desired length.
> For every line segment L, there is a circle which has L as a radius.
> All right angles are congruent each other.
> Suppose we have a ponit P, and a line L not containing P. Is there exactly one line through P that is parallel to L? No, for a very simple reason: in spherical geometry, there are no such things as parallel lines. Any two great circles on a sphere must intersect. Any great circle C cuts the sphere’s surface into two equal parts, each one of which has the same area; call this are A. Now, suppose another great circle, C’, is parallel to C. Since it doesn’t intersect with C, it must be entirely enclosed on one side or the other of C, on one of those two area – A half spheres. But this means that the area enclosed by C’ is les than A, impossible, since every great circle encloes area exactly A.
> There is a saying from Chief Justice John Roberts. “Jundges and justices are servants of the law, not the other way ground. Judges are like umpires. Umpires don’t make the rules; they apply them. The role of an umpire and a judge is critical. They make sure everybody plays by rules. But it is a limited role. Nobody ever wen to a ball game to see the umpire.”
> Most of the mathematicians Jordan Ellenberg work with now weren’t ace mathletes at thirteen; they developed their abilities and talents on a different time scale. There is always somebody ahead of you.
> There is a saying of Mark Twain. “It takes a thousand men to invent a telegraph, or a steam engine, or a phonograph, or a telephone or any other important thing – and the last man gets the credit and we forget the others.”
> Condorcet was the first mathematician who merged politics with mathematics. He was captured and arrested. Two days later he was found dead.
> Math gives us a way of being unsure in a principled way: not just throing up our hands and saying “huh,” but rather making a firm assertion: “I’m not sure, this is why I’m not sure, and this is roughly how not-sure I am.” Or even more: “I’m unsure, and you should be too.”
> Mr. Silver is great. He was willing to talk about uncertainty, willing to treat uncertainity not as a sign of weakness but as a real thing in the world, a thing that can be studied with scientific rigor and employed to good effect.
[Silver knew that Obama would win.]
State / Obama Win Probability / Expected Wrong Answer
OR 99% .01
NM 97% .03
MN 97% .03
MI 98% .02
PA 94% .06
WI 86% .14
NV 78% .22
OH 75% .25
NH 69% .31
IA 68% .32
CO 57% .43
VA 54% .46
FL 35% .35
NC 19% .19
MO 2% .02
AZ 3% .03
MT 2% .02
Mr. Silver got all fifty right. Mr. Silver, instead of saying who was going to win, he reported what he thought the chances were. Instead of syaing how many electoral votes Obama was going to win, he presented a probability distribution; say, Obama had a 67% chance of getting the 270 electoral votes he needed for reelection, a 44% chance of breaking 300, a 21 chance of getting to 330, and so on.
> One criticism of Silver to which Jordan Ellenberg is sympathetic is that it’s misleading to make statements like “Obama has a 73.1% chance of winning as of today.” The decimal suggests a precision of measurement that’s probably not there; you don’t want to say that something meaningful has happened if his model gives 73.1% one day and 73.0% the next. This is very impressively precise-looking but not correct.
> In election, a lot of factors can change the result.
> Jordan Ellenberge spent 8 years for this book.
This book is the most difficult book I’ve have ever read.
I turly appreciate Jordan Ellenberg for great insights.
I Alex Kim want to have a connection with him.
God Bless !!!