The thing of it is, he makes some really good points with this post. A lot of the things we learn in high school took mathematicians **centuries** to come to terms with. The concept of a complex or imaginary number, *i* = sqrt(-1), wasn’t really accepted until the 18th century (why do you think they’re called “imaginary numbers,” after all?). So don’t dispair if it takes you a little while to understand something in math class. Mathematicians of old **died **before they could come to terms with it!

Filed under: Musing Tagged: Comic, Pessimism ]]>

Every so often, the news media becomes all abuzz when a particular lottery jackpot starts to grow really large. Right now is one of those times, with no winner on Saturday putting the jackpot for Wednesday’s drawing at around $1.3 Billion, the largest lottery jackpot in US History.

My students sometimes ask me, as a math teacher and a guy who “knows numbers,” whether I play the lottery. Usually I just smile and tell them I buy the occasional scratch ticket for the fun of it, but almost never anything beyond that. It would require a “special occasion” or a “huge jackpot” for me to consider buying one.

This certainly seems like one of those special occasions.

To understand how to approach this question from a math standpoint, we first need to understand the probability of winning.

The Powerball is a multi-state lottery game run in 44 states as well as Washington, DC; Puerto Rico; and the US Virgin Islands. Only Alabama, Alaska, Hawaii, Mississippi, Nevada, and Utah don’t run the lottery in their state.

Purchasing a $2 ticket involves choosing six numbers, five from a set of 1-69 and 1 from a set of 1-26 (called the Powerball). Only if all six of your numbers come up will you be able to win the full Jackpot prize, though there are lesser prizes for matching smaller quantities of numbers. Helpfully, Powerball has a list of prizes and odds on their website, but let’s take a moment to understand how they are calculated

On the high end, the jackpot requires your five numbers, plus the Powerball, to match those selected on the drawing. Assuming that the drawing is done completely at random, then every possible combination of these 5+1 numbers have the same chance of being chosen. So how many combinations are there?

For the first number you pick, there are 69 choices. You can’t pick the same number again, so there are 68 choices for your second number, then 67 choices for the third, 66 choices for the second, and 65 choices for the last. This would suggest that there are 69*68*67*66*65 = 1,348,621,560 ways to pick your first five numbers, and that gets multiplied by the 26 choices for the Powerball to give an overall number of possibilities of 35,064,160,560.

But that number isn’t correct. The calculation above is known as a **permutation**, which is a way of counting outcomes assuming that different orders of the same numbers are considered different. Permutations are useful when analyzing how many batting lineups of nine players from a team of 37 are possible, or when asking how many ways you can arrange your five family photographs on a shelf. But that’s not what we need here, because the order that the numbers are selected in is irrelevant. What we need to use is a **combination**, which counts outcomes in a similar way, but treats every different arrangement of the same five numbers as the same outcome.

To get a combination, we divide the 1,348,621,560 figure above by 120, the number of ways to arrange five numbers. This gives a total number of outcomes for the first five numbers as 11,238,513, and a total number of possible tickets including the Powerball as 292,201,338. So the probability that you’ll win jackpot is 1 in 292,201,338, which you’ll notice is the exact probability listed on their website. This probability is ridiculously small. Play 50 tickets a day every day, and this probability suggests you will still only win once every 16,000 years (and it’s actually worse than that, since the Powerball drawings only happen twice a week).

Incidentally, this also means that buying all possible ticket combinations – and therefore guaranteeing that you have the winner – would cost $584,402,676. In 1992, an Australian investment firm attempted to do this in the Virginia state lottery, buying 5 million of the possible 7 million combinations. This would still be a bad play, however. You might need to split the pot with another winner or winners, and lottery winnings are federally taxed by up to 25%.

But the Jackpot isn’t the only prize. Look at the bottom end. You win $4 if you only match the red Powerball, which as we said you have a 1 in 26 chance of doing. So why are the odds listed as 1 in 38.32? As the FAQ say, that figure is not **just** the probability of matching the red Powerball, but the probability of matching the red Powerball **and none of the other numbers**. Match at least one other number and your payout is different. To completely miss all five other numbers is a 7624512/11238513, or about 68% chance. Multiply that probability by the 1/26 chance of matching the Powerball, and you’ll get the 1 in 38.32 probability they have listed there.

Now that we have a sense for the probability winning, what do we do that information? How can we break this down and understand the statistical payoff for playing this game?

Consider again buying every possible ticket. One of them is guaranteed to be the jackpot, which right now is projected to be $1.3 Billion. Another 25 of them will have all five numbers match, but not the Powerball, winning the $1,000,000 second place prize. Increasingly more will win the lower prizes, with 7,624,512 having the correct Powerball, but none of the other numbers matching, to win the lowest prize (and, for what its worth, more than 280 million worthless scraps of paper)

If we took the total amount of money won across all winning tickets, subtracted the $2 cost for all 292 million tickets purchased, and divided by the number of tickets purchased, we’d get an average payout per ticket bought. This average is called the expected value and is a reasonably good measure of the value of playing a game (ignoring the business about sharing jackpots and taxes and all of that). We can more conveniently calculate the expected value by merely multiplying each outcome by its respective probability. The table below shows just that, also adjusting each prize for the $2 cost for the ticket.

This looks great! The expected value per ticket is $2.77, suggesting a positive outcome. But remember, we’re ignoring a lot here, taxes and the possibility of splitting the pot. One thing not even mentioned: if you want all $1.3 billion, you will have to wait over a period of 30 years, as the jackpot is only paid out over the long term. If you want the whole amount as cash up front, the actual payout is $806 million. How does that affect the expected value?

The expected value is still positive, but now less than half of what it was before. But this still isn’t taking taxes into consideration. USA Mega Jackpot Analysis breaks down tax rules for lottery winnings state-by-state, and New York has the highest state tax rate of any other state at 8.82%. It projects the after-tax total take-home amount of the lump-sum payout to be $533,410,800. All the other lesser prizes will be similarly affected. How does that affect our expected value?

As you can see, even after taking the lesser lump-sum value of the prize instead of the higher long-term payout, and after factoring in taxes, the expected value is still positive (though just barely).

In short, yes. For perhaps the first time in the history of the lottery, a single-winner would statistically expect a positive return on their “investment”

Then again, If you have to split the pot, the top jackpot value will get cut into equally sized pieces, which will send the expected value into the negatives (-$0.85 if split two ways, -$1.15 if split three ways, -$1.30 if split four ways). On the other hand, the value of the jackpot does depend on how many tickets have been bought, and with all the media coverage this record-breaking jackpot is getting, it wouldn’t be surprising to see the jackpot push closer to $1.4 billion by Wednesday.

I’ll see you in line for a ticket.

Filed under: Musing Tagged: Expected Value, Lottery, Powerball ]]>

The gist is this: Say you need to have a major operation done and there are two hospitals in your town where you could have it. You’re worried about post-surgery complications, so you do some research into the hospitals and find that in the past year, patients at the larger hospital suffered post-surgery complications in 130 out of 1000 cases, and patients at the smaller hospital suffered complications in only 30 out of 300. Based on these results, it looks like the smaller hospital is the better bet: only 10% of patients had complications after surgery there versus 13% at the larger hospital.

However, not all surgeries have the same rate of complications. Relatively minor surgeries are less invasive and would probably result in a lower complication rate. With that in mind, you look further at the data and find that, at the large hospital, 120 out of the 800 major surgery patients experienced complications compared to 10 out of 200 minor surgery patients, and at the small hospital, 10 of the 50 major surgery patients suffered complications compared to 20 out of 250 minor surgery patients. In other words, broken down by type of surgery, the complication rates at the large hospital were 15%/5% for major/minor surgeries while the small hospital saw a rates of 20%/8%. We see now that the larger hospital has a lower rate of complication across the board, regardless of the type of procedure done.

So why the different conclusion? It has to do with **how many** of both types of procedures the hospitals did. The vast majority of the larger hospital’s 1000 surgeries in the last year were major surgeries, which have higher complication rates across the board. The majority of the smaller hospital’s 300 surgeries were more minor procedures, which generally have lower rates of complication. As a result of this imbalance, the overall, pooled complication rates for the two hospitals are biased: the larger hospital towards a higher rate and the smaller hospital towards a lower rate. So it only **appears** that the smaller hospital has a lower complication rate because most of the surgeries performed there are less likely to have complications.

Check out this website for another explanation of Simpson’s Paradox, as well as some clever interactive animations that demonstrate how and why it can arise. It’s an important lesson as consumers of data and statistics: while the saying may go “Less is More,” when it comes to how much detail to include in your research, sometimes less is wrong.

Filed under: Algebra 1, AP Statistics, Fractals & Chaos, Musing Tagged: Paradox, Statistics ]]>

- Sweet Number Pi – Pi music video
- One Million Digits of Pi – can you memorize them all?
- Official Guinness World Record for Most Memorized Digits of Pi – the record is 67,890 places!
- Search for Your Birthday in Pi – mine starts at the 2,373,070th decimal place!
- The Tau Manifesto – Pi is probably not correctly defined and should be twice the value as it is. Many mathematicians call this number “tau” and there is a convincing argument to be made about their point!
- Other Pi Day websites – PiDay.org, PiZone.com

Filed under: Musing Tagged: Pi Day ]]>

Prevents us from discovering and developing young talent. In the interest of maintaining rigor, we’re actually depleting our pool of brainpower.

This post is not to explore the virtues or flaws with Professor Hacker’s arguments, but to point out what many bloggers have recently observed: that the New York Times answered their own question last week in an article about Sony’s controversial movie * The Interview*.

The movie in question, you may have heard, was pulled from theatrical release on the 25th after the studio’s computer network was hacked and threats were made against theaters showing the Seth Rogan/James Franco comedy that depicts the two actors as journalists asked by the CIA to use an upcoming interview with North Korean president Kim Jong Un as an opportunity to assassinate the dictatorial leader. After public outcry from major Hollywood figures and even president Obama, Sony released the film in independent theaters and online. The December 28th NYT article discusses the amount of money Sony earned off of online sales and rentals, but observes that Sony “did not say” how much of the $15 million revenue was from each source (sales vs. rentals).

It would appear that Algebra was not reporter Michael Cieply’s strongest subject either, as there is enough information in this article to set up and solve a simple system of equations to answer that exact question.

Let *s* = the number of $15 sales Sony made and *r* = the number of $6 digital rentals. From the $15 million headline, we can write the equation 15*s* + 6*r* = 15 000 000.

The second paragraph also tells us that there were about two million transactions overall. Therefore, we can make the second equation *s* + *r* = 2 000 000.

Solving this system of equations is a matter any 9th grader can do:

Multiply the second equation by -6 and add vertically

Substitute that value of *s* back into original equation, and you get:

So there you have it, New York Times. With about 2 minutes of high school-level algebra, we can see that * The Interview* saw about 300,000 downloads and 1.7 million rentals in its first four days. Maybe you should employ more ninth-graders…

Filed under: Musing Tagged: New York Times, Sony, The Interview ]]>

For another moment of bizarre, check out the “Also Viewed” section.

Filed under: Musing ]]>

For an awesome example of this, check out Spurious Correlations, a website that takes real data and finds ridiculous correlations between them. For example, did you know that the marriage rate in Kentucky can be a very strong predictor of the number of people who drown after falling out of a fishing boat? Or that the United States decrease in oil imports from Norway seems to cause fewer drivers to die in a collision with a railway train? Or that there’s a clear link to the precipitation rate in Tompkins County and the number of trip/slip related deaths in male Texans?

You can try and find your own correlations as well. If you find something good, post it here!

Filed under: Musing ]]>

Filed under: Musing ]]>

My general recommendation about reading these news articles is, when in doubt, go read the source. Don’t rely on other people to do your thinking for you. Go and seek out the information you need and make your own conclusions!

Filed under: Musing ]]>

The goal of the game is simple: Get a tile of value 2048. The controls are also simple: press an arrow key and every tile that can move in that direction, will. If two tiles of the same value are next to each other, they’ll combine to one tile double that value. Also, with every move, a two or a four will appear at random in a free spot on the board. Seems easy, right?

Well keep in mind that to get a 2048 tile, you’ll need to create and combine two 1024 tiles. To get those two 1024 tiles, you’ll need four 512 tiles, which require eight 256 tiles, which require sixteen 128 tiles. All of these numbers are powers of two, which are the key to the concept of binary numbers, which at the most basic and fundamental level is how computers operate. A binary number is one of base two, in the same way that a “conventional,” decimal number is base 10. Consider the number 2048. You might remember from elementary school that this could be thought of as two 1000’s, zero 100’s, four 10’s, and eight 1’s. Those numbers – 1000, 100, 10, and 1 – are all powers of 10 (10^3, 10^2, 10^1, and 10^0, respectively). A number written in binary uses powers of 2, meaning there is a 2^0 = 1’s place, a 2^1 = 2’s place, a 2^2 = 4’s place, and so on.

Moreover, just as any one place value in a decimal number could be occupied by a digit from 0-9, giving you ten options, a place value in a binary number can only be occupied by two digits: 0 and 1. To write a decimal number like 459 in binary, you first need to figure out how to “assemble” the number using powers of two. The biggest power of two that fits is 256 (2^8). That leaves 203 left, meaning 128 (2^7) fits also. Subtracting that leaves 75, meaning we can take away 64 (2^6). This leaves only 11 left, from which we can subtract 8 (2^3), then 2 (2^1), then only 1 (2^0) remaining. So the decimal number 459 can be rewritten as 11100111. Taken in the other direction, the binary number 110101 would be interpreted as one 1, one 4, one 16, and one 32, giving a decimal number of 53.

What’s important to note is that 53 and 110101 are referring to the same quantity; they are just different ways of representing that quantity. It’s the same way as how “the cat,” “el gato,” and “l’chat” all refer to the same animal. Thinking of decimal and binary as different languages for numbers is actually a great analogy, because binary is how computers think of numbers. The reason why has to do with how computers are made. The circuits on the motherboard, inside the processor, and all throughout your computer are essentially tiny wires. At any instant, the wire either has an active electrical charge running through it or it doesn’t. If the wire is “on,” it is considered a 1. If it is “off,” it is considered a 0. The sender on one end of the wire will turn the current on and off extremely quickly in a manner much like morse code, and the receiver on the other end of the wire will interpret the rapid fire of 1’s and 0’s as binary numbers that can be interpreted in any number of ways.

So far, the best I’ve been able to get in the game is a pair of 256 tiles that I wasn’t able to combine before blocking myself off, so my high score is only 3180. Think you can beat it?

Filed under: Musing ]]>