## GREAT EXPECTATIONS

It all started with a packaging error on some soy burgers I bought.  At home, I discovered a slit in the bottom of two that I didn’t see when I bought them.  Who knows how long the contents had been exposed to air?  How often do you check the packaging when you buy something?  For me, not often enough.

I returned to the store and told the manager, who was shocked.  I said I was going to look at the other packages they had to see if they, too, were defective.  When I checked the box that had the identical soy, sure enough, the two remaining packages in it had rips.  I took them back to the manager’s counter.

I had to wait, however, as I watched a woman with bleached blonde hair, maybe mid 40s, and a young man, who might well have been her son, discuss lottery tickets.

“We can’t afford too many,” she said.  “We need \$20 for xxxx.”  I couldn’t hear what “xxxx” was.

I was stunned, as I watched them purchase six lottery tickets, some sort of scratch type, for \$60, which the woman counted out using twenties, fives and ones.  It might have been a Raffle. You can’t win if you don’t play, right?  That was the catch phrase in Arizona.  You can’t lose if you don’t play would have been more accurate, and I can prove it.

Let us look at the expected value, which anybody playing the lottery should understand.  We ought to teach this in schools. If the expected value of something done is positive, over the long run it will be successful; if negative, over the long run, it will not be.

Here’s an example.  At a roulette table, you plunk down \$1 on a roll of two dice.  If they come up double 3s, you get a 30-fold return—\$30.  Well, it isn’t 30 fold, because you paid \$1 to begin with.  It is 29-fold.  Don’t laugh, casinos get rich on these “minor points.”  You have a 1 in 36 probability that the dice will both be 3: 1 in 6 for each, and we multiply the probabilities when one doesn’t affect the other, a term called  independence. The probability of winning \$30 is (1/36), and the probability of losing \$1 is (35/36).  If you multiply these and then add, you get +\$(30/36)-\$(35/36)=\$(-5/36).  That fraction comes out to about MINUS 14 cents per dollar bet.  Are there winners?  Yes.  The consistent winner is the casino.  Not only is it consistent, they can predict very accurately how much they will make, because in the long run (not a player’s time horizon), the casino will make 14 cents on each dollar wagered for that bet.

What I saw for a \$10 Raffle ticket:

1 in 250 chance of winning \$100

1 in 25,000 chance of winning \$20,000

1 in 250,000 chance of winning \$1,000,000

The exact calculations are more complicated, but the simple way works just fine.

You spend \$10, and you have a 1/250 chance to win \$100, an expected value of \$100/250 or \$0.40, 40 cents.  You also have a 1/25,000 probability of winning \$20,000, and that is \$0.80, so the expected return of both is \$1.20. Add to that the 1 in 250,000 chance of winning \$1 million, \$1,000,000/250,000=\$4.  The total expected return on \$10 spent is \$5.20, give or take.  That’s the plus.  The minus is \$10(249/250)=\$9.96.  The expected value is \$5.20-\$9.96=  MINUS \$4.76.  The pair spent \$60. If we looked at a lot of people buying, all of them, they would on average lose about \$28.50.  One of those \$20s that one comes in to the store with, and nearly a ten spot as well, won’t be seen again.  Do that weekly for a year, one will lose nearly \$1500 on average.

The lottery is very coy about posting long lists of numbers that won \$100.  Sounds like a great deal, except that first, they really won \$90, because they paid \$10 for the ticket.  Secondly, the list of those who lost is about 120 times longer.  We don’t see that one.

Here in Oregon, 93% of the money spent on the lottery goes to payouts.  Sounds good, except that the big payouts go to very few people.  BUT A LOT OF PEOPLE WIN A LITTLE, LIKE \$100, AND THAT’S A HUGE PROBLEM.  This is called variable ratio reinforcement.  Winning occasionally keeps one playing.  Never winning at all causes one to quit sooner.

There is a very strong correlation between low income and high use of the lottery. Why do people play?  Answer: they see this as the only way out of poverty.  The probability is exceedingly high, however, that they will only go deeper into poverty. The lottery is a regressive tax levied on those who can least afford it.  The lottery steals from those who don’t understand math, probability, or how our brain can lie to us.  When the money goes to education, a noble cause, it is being paid for by those who have the least money: the very poor spend 9% of their income on the lottery.  If one makes \$13,000 a year, that is nearly \$1100 they spend on the lottery.  If one has difficulty making ends meet, this is going to push them over the edge—with high probability—and I can define that probability exactly.

I never forgot what my statistics advisor said about expected values:  “If it is positive, I will beg, borrow, steal every dollar I have to play.”

It becomes absolutely certain at some point that somebody will win Powerball.  We can predict that as well.  If there is a Powerball with a probability of winning equal to 1 in 110 million (roughly equivalent to your guessing correctly a minute I choose between the signing of the Declaration of Independence in 1776 and now), and 330 million play the Powerball, the expected value of winners is 3.  Three people are expected to win; the probability of exactly 3 is about 22%; the probability that between 1 and 5 will win is nearly 7 in 8, and the probability nobody will win is about 5%, quite small.  In other words, one can make remarkably accurate probabilistic statements what is going to happen.  Did you guess my minute? (It was 3:32-3:33 a.m. 15 August 1846).

I’m lucky; I live comfortably.  Still, I bend down to pick up a dime or a penny if I see one.  If PetsMart gives me a \$3 coupon for doing a survey, I will do it.  If REI gives me a dividend, and I am planning to buy something, I will buy it with the dividend and furthermore try to get it when the item is on sale.  I use coupons when I shop, I comparison shop, I don’t drive 30 miles for cheaper gas, because it’s more expensive to do so, I pay my credit cards off every month, I try not to get a tax refund, because it means I loaned the government money, and I DON’T PLAY THE LOTTERY.

But occasionally, I do silly things with money.  The store agreed that the ripped packages I returned, and the two others in that I found that were also ripped, needed to be removed.  I found two good packages, but they were not on sale, so I actually had to pay to replace them.  I paid \$4.40 for doing the store and any customer who bought those a favor.  Small price for what I learned about the lottery.

The expected value for doing a good deed was negative.  In the long run…

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