## A good first step in cyber self defense

The recent data breach at Equifax makes it clear that it is critical to use strong passwords for all online accounts. According to Equifax, the data breach happened between mid-May and July. The hack was discovered on July 29, but Equifax did not inform the public until September 7. Sensitive personal information such as Social Security numbers, date of birth, addresses was stolen from Equifax, affecting one in two adults in the United States (see here).

How to create strong passwords? Obviously the longer a password, the stronger it is. The issue is that of laziness. There may be a long list of accounts, e.g. financial accounts (banks and credit cards), social media accounts and other accounts. There may be ten or more such accounts to manage. Some people get lazy and use the same password. Using different passwords that are long enough and random enough across many accounts would indeed pose a huge management challenge – how to remember, store and update the passwords.

It turns out that it is not difficult to create strong passwords that are easy to remember. The idea is to come up with a phrase that is memorable only to you. For example, a college student may come up with the following phrase.

My weakest subject is Chemistry. I will go to Tutoring Center 5 days a week this semester to get Help!

The resulting password would be “MwsiC.IwgtTC5dawtstgH!“. This is a 22-character password that includes upper case letters, lower case letters, numeric characters and special characters. Another plus is that it contains no dictionary words. If the information about chemistry subject is important and meaningful to the creator of the password, then it is easy to remember.

The string MwsiC.IwgtTC5dawtstgH! appears random. Yet there is a high level piece of information behind it that is known to no one except the creator of the passwords.

The college student in question can change the password by using another memorable phrase after the semester is over. So this idea is flexible and the possible pool of passwords is limitless.

For each account, create a memorable phrase associated with that account. Managing these passwords still requires effort. However the information to remember is at a high level (and memorable and personally meaningful). It is not about memorizing a random string of characters. In light of the Equifax data breach, the effort is the least we can do to help defend ourselves.

Any discussion of safe and strong passwords is a good pivot to talking about large numbers. Having an appreciation of large numbers help us appreciate the passwords such as MwsiC.IwgtTC5dawtstgH!.

For example, how many possible 22-character strings are there? To get a sense of how big this number is, let’s assume that the 22-character string consists of only lower case English letters. Then there would be $26^{22}$ possible strings. How big is this number? It is $1.3474 \times 10^{31}$. To simplify, let’s say it is $1 \times 10^{31}$, the number 1 followed by 31 zeros. Note that a billion is one followed by 9 zeros. A trillion is one followed by 12 zeros.

The number $1 \times 10^{24}$ only includes lower case letters. If we include upper case letters as well as numeric characters and special characters ($, ?, ! etc), then the universe of potential passwords is greatly expanded. To appreciate how big $1 \times 10^{24}$ is, let’s compare it with the age of the universe, which is about $13.8 \times 10^{9}$ years (13.8 billion years). Converted to seconds, the age of the universe is approximately $4.35 \times 10^{17}$ seconds, which is less than $1 \times 10^{24}$. Guessing at the password at the rate of one per second, the entire age of the universe is not enough time to cover the possible choices within the number $1 \times 10^{24}$. This is assuming upper case letters and numeric and special characters are not in the mix! It is believed that the sun can burn for another 5 billion years. So guessing at the password at a fast rate would mean that there is not enough time to cover all the possible choices. Using the “memorable phrase” approach for password management is a good first step in cyber self defense. This approach can help keep your bank accounts safe. So it is a good first step in financial self defense as well. Here’s a peculiar way to find strong passwords. This scheme is to produce 26-letter passwords such that every letter is known and is fixed! In fact, the first letter of the password is the first letter in the English alphabets, the second letter of the password is the second letter of the English alphabets and so on. The length of the password is long but every letter is fixed. This scheme is discussed in this blog post. This universe of passwords is not as big as the one for the 22-character passwords discussed above. But it is a big enough collection of possibilities that it is all but impossible to hack without computer help. There are 67,108,864 many different possibilities (over 67 million). How does this scheme work? Why is it that every letter is known but the passwords can be strong? Curious? Think about it or go to this blog post. This particular scheme is a way to learn the concept of binomial distribution. Any one who understands this scheme understands binomial distribution. $\text{ }$ $\text{ }$ $\text{ }$ $\copyright$ 2017 – Dan Ma Advertisements ## Wheat and chessboard problem The wheat and chessboard problem is usually described as placing grain of wheat (or rice) in an 8 x 8 chessboard that is like the following. However, a more colorful description of the problem is: Sessa, an ancient Indian minister, had made a brilliant invention. So the ruler wanted to give him a prize. He could have anything he wished, the ruler said. Sessa requested that one grain of wheat be put in the first square of an 8 by 8 chessboard. Two grains of wheat in the second square. Four grains of wheat in the third square and so on. In general, the number of grains of wheat in one square is always the double of the previous square. The ruler laughed it off as an insignificant prize for Sessa’s achievement and the ruler granted his wish. Later the court treasurers reported that the entire grain reserve of the kingdom was not sufficient to fulfill the request! Basically Sessa was asking for the ruler’s kingdom and more! The problem appears intuitively simple. There are only 64 squares on the chessboard. Surely the ruler of a kingdom would be able to fill the 64 squares with grains in the manner that was requested. Upon closer examination, the growth in grains of wheat is actually phenomenal. The total number of grains needed would be 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + … (all the way to the 64th term) which turns out to be 18,446,744,073,709,551,615. This would be over 18 Quintillion grains of wheat! Or over 18 million times 1 trillion grains of wheat! This story with the Indian minister Sessa is an excellent illustration of exponential grow. When something is growing according to a multiplicative constant (e.g. doubling, tripling) at each step, the quantity can get to be out of control pretty fast. For example, when bacteria count doubles in every minute, the population would be 18 quintillion in just one hour. Note that each term in the above sum is a power of 2. $2^0+2^1+2^1+2^3+2^4+\cdots+2^{63}$ In the wheat and chessboard problem, the base of the exponentiation is 2, hence the doubling in each step. If something is growing in a linear fashion, the increase in each step is an addition of a constant and not the multiplying of a constant. Say, you put one grain of wheat in the first square, 101 grains of wheat in the second square, 201 grains of wheat in the third square and so on. Each square would have 100 more grains than the previous square. Then it would be easy for the ruler to fulfill this request. 1 + 101 + 201 + 301 + … + 6301 = 201,664 The number 201,664 is surely doable. For example, assuming that the population of the kingdom is 200,000, if everyone in the kingdom skips one meal, there would be more than enough grains of wheat to fulfill this request. This is actually a cruel way to treat the citizens. But it is just an illustration that 201,664 is a tractable problem. The number 201,664 surely is a far cry from 18 qunintillion! So exponential growth can dwarf linear growth, depending on the multiplier, by as much as a factor of 18 quintillion. The exponential growth in the wheat and chessboard problem is also illustrated in the game called Tower of Hanoi. The following shows a game set. The exponential story is discussed in this blog post in a companion blog. $\copyright$ 2017 – Dan Ma ## Playing the roulette wheel of (mis)fortune It is well known that playing casino games can lead to ruin (at least financial ruin) if the gambler is not careful (e.g. putting all or a substantial portion of his/her wealth on the line with the goal of beating the casino). There are two ways to prove that such notion is foolish and reckless for anyone that needs to be convinced. One is to through empirical data and the other is through mathematics. Both ways show that the fate of the gambler is indeed grim. The empirical data can be obtained in two ways. One is bringing real money into a real casino to gamble. The other is computer simulation. The first way is at best an expensive way to collect data and at worse putting your financial well being at risk. Game results can be generated using computer generated random numbers based on the presumed odds of the game in question. Since no real money is involved, it is easy to generate thousands or tens of thousands or more bets (or millions of bets if you wish). It is a cheap way to generate results that mimic making a large number of bets. All one needs is a software program that has a random number generator. Let’s simulate results from a roulette wheel (American roulette). In the American version, the wheel has 38 slots – 0, 00 and the numbers 1 through 36.The two slots 0 and 00 are green and the numbers 1 to 36 are half red and half black. The following shows the roulette wheel and the table layout. Players may choose to place bets on either a single number or a range of numbers, the color red or black, or whether the number is odd or even, or if the numbers are high (19–36) or low (1–18). Let’s focus on the bet on the color red. Note that there are 18 red slots out of 38. The payout on this bet is 1 to 1 (if the ball lands on any of the 20 non-red slots, the player loses one unit and if the ball lands on a red slot, the player wins one unit). The odds for winning (for the player) are 18 to 20. Turning it around, the odds for losing (for the player) are 20 to 18. The odds for winning for the player are not 1 to 1 but the payout is 1 to 1. This means that the house has an edge. Essentially, the green slots 0 and 00 give the edge to the house. These two slots give the player extra two possibilities to win or the house two extra possibilities to win. For example, for 38 bets of$1 each on the color red, the player is expected to win 18 times (gain $18) and lose 20 times (lose$20) with an overall loss of $2. In fact, no matter what the bet is (bet on color, bet on even-odd, bet on a range of numbers or a single number), the payout rule is designed so that the player is, on average, expected to lose 2 bets per 38 bets. Since 2/38 = 0.0526 (5.26%), this gives the house the edge of 5.26%. For a more detailed discussion of the roulette game, see this post from a companion blog. If anyone who still does not believe the math, maybe a computer simulation can help. The simulation is done in Excel, which has functions for generating random numbers that can be very handy. For the purpose at hand, we use the function =RANDBETWEEN(a, b), which gives a random number between a and b. In each turn of the roulette wheel, the ball can land in any one of the 38 slots. So we use =RANDBETWEEN(-1, 36). If the result is -1 or 0, we take that as the result of landing in a green slot (-1 for 00 and 0 for 0). If the result is in 1 to 36, we just take it as the result of landing in the slot with that number. We generate 10,000 random numbers using =RANDBETWEEN(-1, 36). To generate 10,000 values, put the formula in cell A1 and then copy the formula down to cell A10000. Next convert these 10,000 numbers from formulas into values (by copying these 10,000 cells and then pasting onto the same cells as values). The following table shows the results on the first 10 plays. Table 1 – First 10 Simulated Roulette Games (bets on Red with$1 per bet)

Random Number Color Player’s Winning
32 red $1 9 red$1
25 red $1 8 black -$1
1 red $1 36 red$1
13 black -$1 3 red$1
14 red $1 23 red$1

The roulette player is doing well in the first 10 games (8 wins and 2 losses) with $6 in winnings. Here’s the next 10 simulated plays. Table 2 – Next 10 Simulated Roulette Games (bets on Red with$1 per bet)

Random Number Color Player’s Winning
-1 green -$1 18 red$1
26 black -$1 30 red$1
29 black -$1 3 red$1
33 black -$1 1 red$1
21 red $1 29 black -$1

In the second 10 games, the roulette player is just making even with 5 red numbers and 5 non-red numbers. In the first 20 simulated games, the ball lands on 13 red slots and 7 non-red slots for the total winning of $6. So in the first 20 simulated games, the player has an edge, winning$0.30 per bet (6/20 = 0.30). As the simulation progresses, the results for the player keeps getting progressively worse.

Table 3 – Summary of the 10,00 Simulated Roulette Games

First n Games Number of Red Balls Winning Amounts Average Winning per $1 Bet 20 13$6 $0.30 100 57$14 $0.14 1,000 484 -$32 -$0.032 5,000 2,371 -$258 -$0.0516 10,000 4,738 -524 -$0.0524

In the first 100 games, the player comes out ahead winning $14 (or 14 cents per$1 bet). If he or she stops right there, the player will bring home real money. If the player thinks that he or she will continue to have 14% edge in his/her favor, he or she is mistaken. At the end of the simulation, the average winning is negative $0.0524, very close to the 5.26% house edge discussed earlier. One thing that should be pointed out is that in actual playing, the player will have to stop long before 10,000 games for the reason of running out of money. For the players that carry only cash into the casino, he or she will run out of cash. For example, assuming that the player starts with$100, the player will run out of cash at the 1372th simulated game in the simulation discussed above. That is a long series of games (being over 1,000 games long). The point is that the rules are stacked against the player. If someone plays long enough, the result is ruin.

Of course, simulations are random (just as actual games of chance are random). Another simulation will produce different results. But the overall pattern will be the same. We generate a few more simulations of 10,000 games each. The following table shows the results.

Table 4 – A Few More Simulations, 10,000 Games each

Simulation # Number of Red Balls Winning Amounts Average Winning per $1 Bet 1 4,691 -$618 -$0.0618 2 4,727 -$546 -$0.0546 3 4,767 -$466 -$0.0466 4 4,718 -$564 -$0.0564 5 4,640 -$720 -$0.0720 6 4,780 -$440 -$0.0440 7 4,713 -$574 -$0.0574 8 4,807 -$386 -$0.0386 9 4,788 -$424 -$0.0424 10 4,739 -$522 -$0.0522 There is indeed a great deal of fluctuation in the results in table 4. Some have more red balls than the others. The average winnings range from negative 3 cents to negative 7 cents per$1 bet.

Another thing to point out is that the number of red balls in Table 4 gives the empirical winning odds for the red ball bet. Recall that the theoretical winning odds for the bet on red ball are 18 in 20 (18 possibilities in winning for the player vs. 20 possibilities in winning for the house). Since 20/18 = 1.11, the winning odds for the player are 1 in 1.11. For each of the simulations in Table 4, the odds are roughly 1 in 1.11. For example, the odds for the first simulation in Table 4 are 4691 in 5309. This translates to 1 in 1.13 (5309/4691 = 1.13).

When performing a random experiment (e.g. making bets at the roulette table), the individual observations are not predictable. However, the long run average of many independent observations is predictable and stable. This is called the law of large numbers. For example, there is no way one can predict whether a player will lose on any given bet at the roulette table. In any one bet, the player may win or lose. In the first 10 simulated games of roulette described in Table 1, the player does quite well. But in the long run, the player will lose (and the casino wins).

According to the law of large numbers, the average results of a random experiment approach the theoretical average. The theoretical winning for the player is negative 5.26% per unit amount. The simulations show that there is a great deal of fluctuation in the individual simulated games. Table 4 shows that the average winnings in the long run do hover around 5.26%.

Playing games of chance for money can be a great entertainment, but only if playing in such a way that the loss is limited and that the amount of potential loss is money that you feel you can afford to lose. That amount is essentially the cost of the entertainment.

Similar simulations are discussed in this post in the context of the carnival game Chuck-a-Luck.

## Listening to tax scammers in action

April 18 is the tax filing deadline in the United States this year. The months leading up to the tax deadline can be a trying time for some, having to navigate among complicated tax forms and the byzantine tax rules. For those who owe IRS money, there may be the worry of finding the funds to pay taxes. The busy tax filing season can surely test anyone’s number skills. Of course, one can always use tax software or hire a tax preparer for help. It turns out that we should also be on the alert for tax related potential hazards, e.g. tax scams on the phone. For this kind of hazard, tax software or hired tax preparer will be of no help. Being good with numbers is an important skill in the tax season. Being informed and staying alert is important too.

Ever wonder what a tax phone scam sounds like? This article from npr.org has a recording of a real life phone scam. The recording is between an active fraud ring and a presumed victim who in reality is a researcher for Pindrop Security, an Atlanta-based company that investigates phone fraud. The original recording is over an hour long, several segments of which are posted in this article.

In the first recording segment, the “IRS” man told “Emma Lauder”, the covert Pindrop researcher that she owed IRS $1986.73 due to miscalculation of taxes from 2009 to 2014. To give the situation a menacing tone, the supposedly IRS many told Emma that local law enforcement personnel with an arrest warrant will be visiting her house any minute now. Her properties will be seized and she will face federal imprisonment of up to five years. Of course, there will be a way out. The way out is through cash! Not through a check or credit card since this is a federal case (this is said at the end of the first recording). The United States federal government is a cash-only organization! The only channel to send cash to the federal government is either Western Union or MoneyGram! What the scammer said on the phone is amazing and incredulous. Think about it. The size of the budget in 2016 for the United States federal government is over$18 trillion. So the scammer is saying that the federal government can only take cash and the cash can only be sent through one of two channels? If that is the case, Western Union and MoneyGram are the most profitable money senders in the entire planet!

Emma is also told to make the payment to a man named Gabriel Porter in Boston. Is Gabriel Porter the Secretary of Treasury of the United States?

As incredulous as these schemes seem to be, they are effective enough that over 5,000 victims have forked over $26.5 million to such scammers since October 2013, according to this tax scam alert from IRS. If anyone receives such phone call or email, report such calls to IRS immediately (call IRS at 800-829-1040). Remember IRS will not call anyone for delinquent tax bills without first mailing a tax bill (through the postal service). Here’s several bullet points from the alert that are worth repeating. IRS will never: • Call to demand immediate payment, nor will the agency call about taxes owed without first having mailed you a bill. • Demand that you pay taxes without giving you the opportunity to question or appeal the amount they say you owe. • Require you to use a specific payment method for your taxes, such as a prepaid debit card. • Ask for credit or debit card numbers over the phone. • Threaten to bring in local police or other law-enforcement groups to have you arrested for not paying. The mere fact that someone who claims to be from IRS calling you about taxes you owed is enough of a red flag. But those scammers prey on the fear among the potential victims. This is why the recordings posted in the npr article can be a very effective public service announcement. I urge everyone to listen to the recordings. This recording (in several segments) is like an anatomy of a tax scam. You can hear the scare tactics. You can hear for yourself how the scam works. You hear a vivid telling of all the absurd details. Like the scammer telling the victim not to tell anyone. Like you can only wire money to the federal government through Western Union or MoneyGram. They say a picture is worth a thousand words. In this case a phone recording definitely makes the scam feel real. At the end, the conversation between the scammers and the victim turned surreal. Emma, the covert Pindrop researcher, asked for a receipt of the payment. Pressing for an email receipt of the payment, she was given a confirmation number instead. The letters in the confirmation numbers form a slang term for the female genital. But the insult is not clear at first since it is a foreign slang. As for the actual receipt, she was told to look into her toilet. This is downright creepy and abusive. The recording in this npr article is a good tool to publicize such fraud schemes. Tax scams can happen any time of year, not just at tax time. Here’s more information from IRS on tax scams. ## Siege of Leningrad A picture can tell a thousand words. Sometimes a number can give a good picture. In the example discussed here, a picture and a number can be combined to make history come alive. I recently watched a World War II news reels on YouTube called FRONTLINE WWII: Germans Advance into Russia (720p). The video describes the operation called Operation Barbarossa, which was the code name for Nazi Germany’s invasion of the Soviet Union during World War II and was launched on Sunday 22 June 1941. One of the three strategic objectives for the operation was to capture the city of Leningrad, now called Saint Petersburg. The German laid siege to the city for 872 days from September 8, 1941 to January 27, 1944. Early on in the war, the Germans believed that Leningrad would be taken easily. In fact, it has been reported that Adolf Hitler was so confident of capturing Leningrad that he had invitations printed with the victory celebration party to be held in the city’s Hotel Astoria. Later Hitler made the strategic decision to divert resources to other fronts. The plan for Leningrad was changed from direct attack and capture to a siege with the goal of starving the city into submission. The siege of Leningrad was one of the longest and most destructive sieges in world history. The destructive impact on the city is detailed in the Wikepedia entry on the siege of Leningrad and in countless other sources. When I watched the YouTube video, one number stands out. During the siege, each soldier or worker doing critical work received 8 ounces of bread a day (and nothing else). The other residents of the city received daily ration of 4 ounces of bread. It did not matter if a resident was young or old, healthy or sick. If the person was not fighting, he or she only had 4 ounces of bread per day for sustenance. To get an idea how much food is an 8-ounce piece of bread, the following is a picture of a loaf of bread that is found in any grocery store in United States. The loaf pictured is a 24-ounce loaf. That means that the a Soviet soldier defending Leningrad received about one third of a loaf of bread for an entire day. Here’s the math: 24 oz x 1/3 = 8 oz. But that is only in terms of weight. The quality of the bread that a soldier received could not be compared with the loaf pictured above. The bread during the siege was made up of sawdust and other inedible ingredients (50 to 60%). The loaf pictured above has 18 slices (you can actually count the slices). The daily ration is then 6 slices of bread for a soldier and 3 slices for everyone else (children and the elderly). Here’s the math: 18 x 1/3 = 6 slices. So a child or an old person subsisted on 3 thin slices of bread that was half saw dust! The picture was indeed grim. The deaths in Leningrad peaked at 100,000 a month in early 1942, mostly from starvation. Due to the lack of fuel, the trolley service ceased to work for most of the siege. Just to get the meager 4-ounce ration of bread, people would need to walk to a distribution kiosk. In a typical winter in Leningrad, the temperature can drop to minus 30 Celsius (minus 22 Fahrenheit). For many people, the walk to a distribution kiosk would be an insurmountable obstacle. The Nazi siege of Leningrad that began in 1941 and ended in 1944 was one of the most gruesome episodes of World War II. Nearly three million people endured it. Altogether, the siege lasted nearly 900 days and resulted in the deaths of more than 1 million civilians. The siege of Leningrad was an epic story of sufferings and destruction and ultimately triumph. If the daily ration of bread piques your interest, there are many places to read more. Here’s some links. __________________________________________ Why Write about Numbers This is a blog about making sense of numbers. The loaf of bread example shows that sometimes it is the other ways round – finding just the right numerical examples to help us understand a complex story or a complex phenomenon. Please feel free to browse the articles in this blog. Here’s are some articles that may be of interest. Here are two posts on number sense. The first one is on stealth price increases. Some manufacturers do not raise prices but give you less. For example, a pack of peanut may have 16 ounces before and now weights 14 ounces but is charged the same price as a 16-oz pack. This post shows how to calculate the price increase. This post is a plug on quantitative literacy after an encounter with a store clerk. I had written on lottery, especially how small the odds are. Buying lottery tickets as entertainment is one thing. Anyone buying them as investment or as quick ways to get rich should know that it will take buying thousands of tickets each week since the time of Christ to have a realistic chance of winning the Powerball lottery. The following are some of the most popular posts in this blog. ## How wealthy are the world’s eight wealthiest people? To find out the total wealth of the top 8 richest people in the world (Bill Gates, Warren Buffet and six other people), we can certainly look up publicly available information on the Internet or from other sources. According to one tabulation based on data from Credit Suisse Global Wealth Data book 2016, the total net worth of these 8 richest men (they are all men) in 2016 is$426.2 billion. Of course, this is a big number. What is a good way to put this number into a proper context? For further meaning of this number, a comparison will be helpful. It turns out that according to the same tabulation, the total wealth of the 3.6 billion people in the poorer half of the current world population is $409 billion. This is a stunning contrast and is a vivid demonstration of the wide gulf between the richest and poorest people in the world. There is no need to use chart and graph to enhance the demonstration. Just the fact 8 individuals own more wealth than the combined wealth of the people in the bottom half of the world’s economy is enough to convince anyone of the disparity between the have and the have not. I came across this report recently in an article from npr.org. It discussed a recent report released by Oxfam International on global wealth inequity. Here’s the press release by Oxfam on this report. Here’s the actual report by Oxfam. Oxfam International is a confederation of 19 organizations working together with partners and local communities in more than 90 countries to combat poverty. The top 8 richest people are, in the of net worth: 1. Bill Gates: America founder of Microsoft (net worth$75 billion).
2. Amancio Ortega: Spanish founder of Inditex which owns the Zara fashion chain (net worth $67 billion). 3. Warren Buffett: American CEO and largest shareholder in Berkshire Hathaway (net worth$60.8 billion).
4. Carlos Slim Helu: Mexican owner of Grupo Carso (net worth: $50 billion). 5. Jeff Bezos: American founder, chairman and chief executive of Amazon (net worth:$45.2 billion).
6. Mark Zuckerberg: American chairman, chief executive officer, and co-founder of Facebook (net worth $44.6 billion). 7. Larry Ellison: American co-founder and CEO of Oracle (net worth$43.6 billion).
8. Michael Bloomberg: American founder, owner and CEO of Bloomberg LP (net worth: $40 billion). The number of the richest people with total wealth equal to the poorer half of the world works like a metric to measure wealth inequality. So it is in some sense a measure of wealth gap. It is not clear if this measure of wealth gap is used by anyone other than Oxfam. But it sure is an effective metric. It gives a vivid demonstration of the wealth disparity between the richest and the poorest. In 2016, “the number of richest people = half of world” metric is 8 as discussed above. In past years, the number was higher. For example, in 2015, the number was 62. Going back further the number got progressively larger. $\displaystyle \begin{array}{rrr} \text{Year} & \text{ } & \text{Number of Richest People} \\ \text{ } & \text{ } & \text{ } \\ 2010 & \text{ } & 388 \\ 2011 & \text{ } & 177 \\ 2012 & \text{ } & 159 \\ 2013 & \text{ } & 92 \\ 2014 & \text{ } & 80 \\ 2015 & \text{ } & 62 \\ 2016 & \text{ } & 8 \end{array}$ The above table is found in the Oxfam press release in 2016 (for the 2015 data). Even in 2010, the contrast with 388 richest people equal half the world is already a clear and lopsided contrast. The trends exhibited in the table show that the wealth gap keeps getting wider. In addition to the 8 richest people equal half the world, the same report also shows that the 1,180 people in the 2016 Forbes list of the world’s richest people own as much wealth as a full 70% of the rest of the world. The average wealth of the 3.6 billion people in the poorer half is$113.6 (=409 / 3.6). On the other hand, the average wealth of the top 8 men is $53.275 billion (= 426.2 / 8). This is again a contrast as skewed as the one mentioned at the beginning (basically 50 billion dollars versus 100 hundred dollars). It will be interesting to know the median wealth of the 3.6 billion people in the poorer half of the world economy. It is likely that the wealth data in the poorer half is skewed as well, meaning that most people have very little (say subsisting on$2 a day) while the top part in the group have wealth in the hundreds or thousands.

Statistics is comparative. There are other comparisons that can be made. For example, how does the total wealth of $426 billion stack up against the GDPs of countries of the world. Of course, we cannot expect the total wealth of Bill Gates and Warren Buffet and 6 others men to equal the GDP of a large economy like the United States. The GDP of the United States is over$18 trillion, which is over 42 times the combined wealth of the 8 richest men. This is still a staggering amount of wealth. Take the combined wealth of the 8 men discussed here, multiply that by 42 and we are approaching the size of the US economy! According to this GDP ranking, $426 billion is about the same as the GDP of Iran, which is about$425 billion. Iran is in the 26th largest economy in the world (according to one of the three ranking in the Wikipedia link). Any way you cut it, these 8 individuals are very wealthy.

## A Periodic Look at the California Lottery

What are the odds of winning the California Lottery? I am talking about the winning of $1 million or more (the kind of winning that is a game changer in one’s personal life). How often are these million-dollar tickets won? Ten months ago I estimated that the odds of winning$1 million or more in the California Lottery were one in 36 million (see Taking another look at the California Lottery). The data were based on data from California Lottery that I obtained in November 2010 (see Shining a light on the California Lottery). Nothing happened in the last ten months indicates that the odds of winning has fundamentally changed.

Just to confirm, I count the number of winning million-dollar winning tickets as of today (August 30, 2011). This is done at the website of the California Lottery. The data are not readily available. I have to search at this site. I count the tickets by searching one county at a time (there are 58 counties in the state). The result: since the inception of the California Lottery in 1985, there are only 257 tickets that paid out \$1 million or more (an increase of 10 winning tickets over 10 months ago). So in its 26-year history, there are only about 260 winning tickets, about 10 per year. The increase of 10 tickets in the last 10 months also confirms the average of 10 winning tickets per year.

With the increase of 10 more winning tickets, the odds are actually a little higher, about one in 36.7 million, but still not fundamentally different from 1 in 36 million.

The mantra of many lotto players is that you have to buy a ticket in order to win. That is so true. You have to get in the game to have a chance to win, even though the chance of winning is infinitesimally small. On average it takes the purchase of about 36 million tickets to support one winning ticket. Still dreaming of winning big?