The Math of the MA State Lottery: Fishy Probability April 26, 2011

The probability of losing on 30 scratch tickets in a row is 1/192,307.  My friend is the luckiest unlucky person ever.

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My brother witnessed me scratch my first ticket.  Though the mechanism of scratching is hardly difficult, I managed to mess up one part of the code by uncovering the prizes for all of my numbers.  When my brother looked over and saw “1 MIL”… well, let’s just say we were both a bit disappointed.  That ticket was my first contribution to Massachusetts’s secret underground revenue stream where there are no checks and balances, just tickets.  Everyone wonders where their tax dollars go and, when we take home just 2/3 of the amount we’re told we make, why our cars still get swallowed by pot holes into the summer.  That being said, public schools are worth every penny I pay in taxes.  But taxes aside, what happens to lottery money?  Is there any system in place to assure that the odds printed on the backs of tickets are accurate? 

 

For my friend’s 30^th birthday, I bought her 30 $1 scratch tickets with the idea she’d win something.  Anything.  The thought barely crossed my mind that all 30 of those tickets would end up in Monday’s recycling pile.  So what did she win?  Nothing.  Clearly printed on the front of each of these 30 tickets was the probability that “one in three is a winner”.  Based on this ratio, she should have won 10 times on 30 tickets.  OK, so maybe probability doesn’t always mirror real life, but can a girl get a win?  When I posed this question to the math blogger Josh Rappaport of mathchat, he gave the following response:

 

Hi ZS, assuming that whether or not one wins or loses on one scratch ticket (what is that, anyhow?) is independent from winning or losing on any other scratch ticket, you treat each event as an independent event. Laws of probability tell us to multiply the various probabilities of independent events. It appears that the probability of [losing] on any particular scratch ticket must be 2/3. So then the probability of [losing] on 30 scratch tickets in a row (if that is what your problem is asking) must be (2/3)^30 = approximately 5.2 x 10^–6, which is about .0000052, or 52 out of 10 million, which boils down to 1 chance out of 192,307.

 

The chance of my friend losing on all 30 tickets, like she did, was 1 in 192,307.  If 192,307 people all got 30 scratch tickets each, just one – my friend – would lose on all 30.  Something seems a bit off in the Massachusetts State lottery.

 

My thoughts here are that scratching a ticket is not truly an independent event, though there are so many tickets printed that it might as well be.  If we were to work this as a dependent probability problem, we’d have to know how many tickets are printed.  So how many are actually printed?  It strikes me as suspicious that the only people who know this figure are the very same people who are in charge of dolling out – or, more accurately, not doling out – the prize money.

 

A lot of people spend more on scratchies than they do on food.  I am not one of them.  The price I spend on food every couple weeks is comfortably higher than the cost of all the scratch tickets I have ever bought.  Still, I sometimes like to test my luck.  At the time of my first ticket, I was living in Southie.  For anyone who knows the area, my apartment was, not unlike many apartments in this area east of downtown, sandwiched between a convenience store and a liquor store, both of which sold scratchies.  Spent tickets littered the streets.  Spent people littered the streets.  It truly was an avenue of broken dreams.  Still, I’d win sometimes.  The $100 I once won somehow felt much more than 1/8 of my rent at the time and I vowed to keep the five crisp $20 bills in a secret place in my apartment.  They were all gone next grocery day. 

 

Buying a scratch ticket now and again is OK for a person who has a steady job, is paid decent money and has been educated on the dangers of gambling by parents who do not gamble.  Scratch tickets comprised a small sliver of my budgeted entertainment money and everyone needs a good adrenaline rush now and again.  But what about my neighbors in Southie, waiting for the bus frantically scratching tickets?  Who is going to stop them from falling into this trap?

 

Moreover, what if the game changes?  Of course there is no real way of verifying my claim, but scratch tickets aren’t paying out like they did five years ago.  Whereas I would win every once in a while, I have not won on a ticket in enough time to make me feel something is wrong.  My rational mind does not conclude that I am unlucky, it tells me there’s something fishy in Denmark.  More specifically, there’s something rank in the Massachusetts State lottery; they changed the rules mid-game and are back ally robbing the Massachusetts working class.

 

The last ticket I scratched – a sleek black $5 number – directed me to a website to learn its odds.  I have searched online for how many tickets are printed but have come up empty.  What is the probability of winning?  Is anyone overseeing that the Massachusetts State lottery is operating fairly?  Where does all this money go?

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overkilling negatives? May 8, 2010

 

I know the ruler seems a bit overkill for a simple subject like adding positives and negatives, but I teach 11th grade in Boston and it’s the biggest stumbling block for even my students taking my advanced algebra class.

 

The problem is that kids are taught a “noun-verb” way of solving problems like “-12 + 7″. They are told to find -12 (noun, static number) and count up 7 spaces (verb, movement) to the right to see what number they land on. This is fine in a classroom with a number line taped to the desk, but it doesn’t teach the kids how to think about the numbers and a lot of kids will get this problem, and ones like it, wrong. It only gets worse with “x + 12 = 7 (solve for x)” or “y + 12x = 7x + 3 (solve for y)”. It’s the same problem over and over again, just disguised.

 

The problem with the number line and the “noun-verb” way of solving is that it’s not the way we think. It’s not even the way we are taught in school to solve these problems. In the Boston 7th grade curriculum is a book called “Accentuate the Negative” where the very first page of text has a caption over a kid’s head that reads something along the lines of “I owe my dad $4. I have -$4″. So this business of “owing” comes into play very early.

 

If I owed you $12 (-12) and I only paid you back 7 (+7), how much would I still owe you? Asked like this, it’s a simple problem. You’d count up from 7 until you got to 12, knowing that the answer would be in “owe”, or negative. In school however, the kids are told to start at -12 and count up 7 spaces. This is completely backwards from how we think.

 

So to get to my ruler…. The ZeroSum ruler allows a kid to find -12, find 7, fold the ruler in half and count the space between the two numbers’ absolute values. This is what we do when we are finding out how much someone owes us, and this is really the way we think. In time, and to answer your question about what a kid would do with numbers beyond -25 and +25, a kid would start to see the relationship between positives and negatives and that if you “owe” more than you “pay” (if the negative is further away from break even (zero) than the positive) then the answer will take a negative sign. But it’s really the space between the absolute values we are counting.

 

 

So, how much do I owe you? April 19, 2010

 

You friend borrows $22 from you.  He pays you back $15 the next day.  How much does he still owe you?  Asked this way, it’s obvious he owes you $7.  But give a kid the problem -22 + 15, and the answer mysteriously becomes, well, mysterious.  

 

WHY?

  

My students can certainly tell me how much I would still owe if I borrowed $22 and paid just $15 back. Like us, they’d probably count up from 15 to get to 22. But give a student the problem “-22 + 15″, and all bets are off.

  

For this number sentence, we are taught in school to find “-22″ on a number line and count to the right 15 spaces to find the number we land on. But this is not what we do in real life to find out how much someone still owes. There is a huge disconnect here.  In real life, we count up from 15 to 22, keeping a tally on our fingers of how many numbers we pass by.  We would never count up 15 from -22 to find how much someone owes us!  It’s no wonder students have difficulty with negative numbers with the way we are taught!

  

To plug my product, the ZeroSum ruler allows a student to count the spaces from 15 to -22 by folding the ruler in half at the pivot and counting from 15 to +22. When the positives are aligned with their negatives, they’re essentially finding the difference between the absolute values of -22 and 15.  This is the way we think and therefore a more natural way to learn.