I taught probability distributions today, and their associated expected values, variances, and standard deviations. It's one of my favorite lessons, so I thought I'd blog about it.
Prior to the lesson, I print off pages for scratch off tickets from the Minnesota State Lottery website. You know, tickets like this:
And the state lottery site is also nice enough to put up the probability of winning and how many of each prize was printed:
So I print off enough pages for everyone or every pair. I used to just give them out randomly, but I find it fun to hype the tickets a little bit. Ohhhh, $100,000 Poker Night? That's a good night! ELECTRIC 8's!!?!? Whoa that sounds dangerous AND FUN. Double Dollar Fortunnnnnnne in the mfing house!!! The TWINS may suck, but you could win big! Worth a MILLION??!?! Are you kidding me?!?!?! I'm never coming to work AGAIN! You know, that sort of thing.
Pause: Those links above will expire; I promise. If you're reading the post later than Summer 2013 you'll need to use your imagination or make up your own.
So I go through the tickets after I've hyped them a little and I tell them that I'll give the ticket page to the first hand I see. I read them off. There are some weird titles like "Scratch Me" that if you deadpan the delivery can get a big laugh. I tell them that if they don't raise their hand I'll assign them one. And nobody wants that. The hype works.
I take the leftovers and pick the one with the smallest number of rows in the distribution table. Today it was MONOPOLY.
If winning up to 27 times doesn't get you excited, I don't know what will.
So everybody's got their ticket, and I've got mine. And the questions are obvious:
"Who's got the best ticket?"
"Which one will win the most money?"
"What's the probability of winning?"
"How many do I need to play on average before I win?"
Those types of questions.
Of course, I have hidden motives. To answer the question, it's helpful to write it as a probability distribution table.
Whoops, we forgot a row. Can anyone think of what we may have forgotten?
At this point it's important to leverage technology, so we put the $ list & probabilities into our calculators. And we take a brief pitstop in the land of cumulative probability distributions. I'm doing mine and they're doing theirs. Everyone has a different ticket, and this is the part where it's helpful to have kids who know how to use their calculators. If you don't know, the "cumsum(" button works very well for this.
Very easy now to know the probability of winning $15 or less, for example.
But that is not the meat of the lesson. Here it comes. Expected value. How much, on average, will you win if you play a lot? Time to define expected value. Note the beautifully well-written mu. I have incredible mus.
We delete L3 and put in a new calculation: L1*L2. Add it all up and that is our expected value. If we play a lot, I will win, on average, $7.00 in the long run. I NEED TO BE PLAYING MONOPOLY; I'M GOING TO MAKE BANK AND QUIT MY JOB!!!
(Groans, boos, and a couple of cheers, as you might expect.)
Turns out I spent $10 on the ticket so I'm actually losing an average of $3 every time I play. Now we go around and share out how much we'd lose if we play their respective games. Interestingly, but not surprisingly, you lose about the same amount for each type of ticket of the same denomination.
Who's ticket is the best? Depends upon how you define "the best"? Scratch Me, a $1 game, could be the best because you lose the least every time you play. Or maybe it's Worth a Million, because you can win a million dollars. "Actually they all suck because you lose money on all of them," says a student.
Life lesson intersects math. That's a win.
Here's where I insert my story about a buddy who uses the lottery to hide money from his wife. I really hope his wife never comes across this blog because she'll know who I am talking about and he'll get into trouble.
He plays $5 or $10 tickets once or twice a week. His wife never notices because the denominations are small and they don't really factor into their budget. But when he hits a winner, that money is all his. Money he can spend on booze, drugs, hookers, whatever. OK, it's not that exotic. Usually, we just get a couple of beers and play a little Golden Tee at Buffalo Wild Wings.
I'd suggest he just stuff $10 into a hidden pillow case every week, but I suppose that's not as fun.
Now I hit them with this. They saw variance for data distributions earlier in the year, but it's a bit different with probability distributions.
AND, we calculate our variances in our calculators without much thought about what it means. I got 120849. What the heck does THAT mean?
We need to do some more splorin. Who's got a $5 ticket? I'll bet yours is less than mine. $2? Less than that. $1? Yeah, much smaller. How about my $30? Is yours over a million? Discuss with your partners what this number might tell us.
We finish boringly with a small note about standard deviation. Yes, it's still the square root of the variance. And it still measures spread. Lesson ends on a whimper.
Overall , although I am sure many out there do something similar, it's a still a lesson that I love; one that brings the math home to real life. It really energizes me when I can do this kind of thing. It put me in just the right mood to teach solving rational equations to Algebra 2 students later in the day, completely devoid of context and without any application to anything any of them will ever do again.