What were the odds? September 30, 2000

We love playing gambling games. Publisher's Clearing House survives due to the propensity of people to believe that Ed McMahon WILL show up on their doorstep. Yet, simply examining the odds of many of these contests reveals the slim chance of winning. That is, if the contest is set up properly. While at CalTech, I was involved in a project that exploited a poorly set up contest. (CalTech, like many schools, has a history of pranks. What is interesting is that one of its best is now used as a principle example of a "hack". Since the UW was involved, you should check out this link in the Jargon Dictionary.)

McDonalds in the LA area was having a "Let's Bust Inflation" contest. Remember this was the mid-70s when inflation was running well over 10% a year. If you were a registered driver you could type your name and address on a piece of paper and put it in a box at participating McDonalds. (It did not even cost the price of a stamp.) And you could enter as many times as you wanted. Each McDonalds would have some small prize winners but the grand prizes, such as a car or a years supply of groceries, would be chosen from all the winners.

So, seeing as how there were not enough women around to divert our attention, a group of us came up with a plan over Winter Break. Nothing in the rules prevented computer printouts. So we took the name of every valid driver living in the house and wrote a short computer program to print out a million or so entries. Not actually an easy task. Remember, this was in the days of punch cards and line printers, so we had to print out our million entries at night, when the Computer Center was not in heavy use. We cut up all the printouts to the proper size, divvied them up and, using a map of every McDonalds in the area, stuffed every entry box. Needless to say, McDonalds was not very happy. We donated most of the prizes to charity, keeping a small amount for parties. (The best quote that came out of this prank was from the main conspirator. When asked by the LA Times what we planned to do with any money won, he said "Parties!! Lots of Parties!!"). So, be sure to check most contests today. They either limit the number of entries or require you to spend money for a stamp.

I, along with several other million people, bought a Lotto ticket last week. I figured it was only a buck, but I was under no illusion that I would win. (Well, I did win $3 so I guess I came out ahead this time.) But there is no reason to buy more than that because the odds really do not change. Spend 5 bucks instead of 1 and your chance of winning goes from 1 in 7 million to 5 in 7 million. That means that you have gone from a 99.999986% chance of losing to a 99.999929% chance.

And I only buy a Lotto ticket when the jackpot has gotten above $7 million. If I bought every number combination, it would cost me $7 million dollars. If the jackpot is only $2 million, I would not break even. Spend $7 million to get $2 million? No way! But if the jackpot is $14 million, I would be a sure winner and make $7 million after recouping my initial investment. Of course, this only works if I am the only player. (There are a few other problems that I will leave as an exercise for the reader.) In real life, others are involved. If 2 of us bought all possible combinations, the state would get $14 million, but we would split any winnings. So then the jackpot would have to be $14 million for us to simply recoup our investment. Not so good.

But the more you play, the better your odds, right? This is correct. So, standard probability analysis would show that, if you played once a week, every week for the next century, you would have a 99.4% chance of picking 4 out of 6 numbers at least once (which would garner you about $550), a 9% chance of picking 5 and a 0.04% chance of hitting the jackpot. So your chance of losing has gone from a virtual certainty of 99.999986% to a much more palatable 99.96%. (Deep sarcasm intended.) A sucker's bet. There are better ways to invest your money. Although 25% of the population doesn't think so.

I love this page. It lists things that are more likely to happen to you than win the lottery. You are 10 times more likely to be dealt a royal flush in poker than win the lottery. You are more likely to die in an automobile accident driving to buy your Lotto ticket than win the lottery. Most state-sponsored lotteries work to obfuscate the actual chances of winning from most of the public. I mean, if the lottery said, pick 2 numbers between 1 and 14 million, would you really think you had a chance of winning?

On the way home last week, I listened to one of the radio personalities interviewing a woman who has written a book detailing what you need to do to win a lottery. She had a system for picking numbers. See first off, you never want the computer to choose it for you. You need to chose your own using her system. One so good she wrote a book about it. Right? I actually started talking back to the radio, listening to this woman spout information that is misleading at best. I mean, if her system is so good, why isn't she using it to win a few Lotto jackpots?

Our Lotto works by picking 6 nonidentical numbers from 1 to 49. The radio host had stated, correctly, that the chance of winning by picking the numbers 1, 2 , 3, 4, 5, 6 was the same as picking any other set, even the winning one. This author stated that this was incorrect and that you would never win using those numbers. Of course, almost no one is going to win using her method, but she ignored that.

Now, this author gave away several of her secrets. She has a database covering every lottery in the world. Using this "extensive" database, she had come up with some important tips for picking a winning number. She said you should have an equal number of even and odd numbers. Well, duh. Assuming the Lotto is actually fair, than randomly choosing numbers between 1 and 49 will give you even numbers just about half the time, and odd just about half the time.

She also had a pseudo-mathmatical rational that disqualified the choice of 1,2,3,4,5,6. She said that the numbers needed to add up to 140. Let's look at this more closely. I may not have all the math down, but I think I can explain this. What is the "average" number? If you randomly choose a number from 1 to 49 100 times, added them all up and divided by 100, what would you get? Intuition would say 25, halfway between 1 and 49, which is correct. I could give you the math but let's keep it simple. Any number actually chosen that is less than 25 will be matched by another number that is greater. At least if enough trials are done.

If the mean number is 25 and 6 numbers are chosen, than the sum will be 150. (Remember, this is the average sum from an infinite number of trials. Any particular trial will have have a sum ranging between 21 and 279.) So, examining an infinite number of lottery drawings will give us an average sum of 150. The fact that this woman proposes 140 simply indicates that not enough trials have been done. There is no method here. No method except to sell books and to make a buck off of the gullibility of people.

The sequence 1,2,3,4,5,6 will be a winner someday, if enough lottery drawings are made. It would just take a long time for that particular number sequence to come up. Roughly 20 million drawings would have to be performed to have a 90% chance of having this sequence occur. I am quite sure that there are not that many entries in this woman's database.

Now, one number WILL be a winner. But your particular number is not very likely. It is kind of like the odds in playing poker. The chance of picking the sequence of any 5 cards before they are dealt is 1 in 300 million. Not very good odds. But, luckily, you don't have to predict what your cards will be beforehand. You examine the cards you are dealt and then play the odds. And someone will win.

Now add on a selection process and your odds improve tremendously. In draw poker, you can examine your cards, discard those that you do not like, and get several more. So, your chances of getting a really good hand increase because you have selected "good" cards from the random set you were dealt. I have seen people get a Royal Flush in draw poker because the odds of adding the necessary cards in a draw are so much lower than simply being deal the ultimate pat hand.

A similar thing happens with natural selection. Most people just do not have a good handle on how natural selection can drive a random process to something that is not so random. Here is another example of how powerful selection can be in improving the odds. Toss 50 coins. You win when you have 50 heads. Now the chance the you will toss 50 heads simultaneously from all 50 coins is something like 1 out 10¹⁵. Not likely. If you had to toss all 50 simultaneously, you would never see this happen. But, what if after each toss, you could select those that were "good". That is, save the ones that are heads and only retoss the ones that had been tails. Just as in draw poker you get to keep the "good" cards and get rid of the "bad" ones. You could get to 50 heads pretty fast that way.

Einstein once said "God does not play dice." He was describing the horrible idea that the probabilistic aspects of quantum mechanics could not be a proper description of the world. Atoms are things, not probability fields describing the possibility that they exist at a particular place. This reliance on probability to describe physics has been one of the greatest paradigm shifts in the field, but also one of the hardest to really understand. Well, a similar shift may be occurring in biology. Probability and statistics are entering the field many of us chose because of the easy math requirements. Things may never be the same. I'll talk more about this next week and introduce a wonderful word, stochastic.