Monthly Archives: September 2011

Are lottery odds incorrect, and your chances drastically worse?

A simple application of elementary probability shows that lottery odds as reported by the operators are incorrect.

Note: I have delayed posting this for a while since the results I come up with seem incorrect. Hopefully someone can respond and tell me where the problem lies.

About a year ago while speaking with my brother Robert on the phone he casually mentioned a joke he made. He said lotteries are so funny since you have to pick the winning number twice in order to win. We both laughed.

But, then later I did a double take, huh? That is true, you have to pick the number set, and then a few days later, the lottery company will also pick a set. If the sets match then you win. Ok, that makes sense. But, if seen this way, the probability value they give for winning couldn’t be correct. Could it? They only give the odds of picking any set, not the winning set. It has to be much harder to win, thus, the probability much lower.

How to compute the Probability?
In probability theory there are rules for combination of events. If the events in an “experiment” are independent, you just multiply the probability values of each: P(A and B) = P(A intersection B) = P(A)P(B).

Further details are on this High School wiki page:

Multiplying probabilities

Probabilities are multiplied together whenever an event occurs in multiple “stages” or “steps.” For example, consider rolling a single die twice; the probability of rolling a 6 both times is calculated by multiplying the probabilities for the individual steps involved. Intuitively, the first step is simply the first roll, and the second step is the second roll. Therefore, the final probability for rolling a 6 twice is as follows:

P(rolling a 6 twice) = P(rolling a 6 the first time) X P(rolling a 6 the second time) = 1/6 X 1/6 = 1/36 approx 2.8%

Similarly, note that the multiplication of probabilities is often associated with the use of the word “and” — whenever we say that some event E is equivalent to all of the events X, Y, and Z occurring, we use multiplication to combine their probabilities (if they are independent).

More info on this wikipedia entry: Probability, Mathematical treatment

Does this apply to the lotteries, like Powerball? There are two events, though separated by days. The consumer, player, picks a set, then later the operator picks there own set. And, they are independent, neither event is dependent on the other. So, the problem, to me, is interpreting the “experiment”. I contend that the whole game, which takes place over a few days is one thing, an experiment, and so the multiplication rule applies.

Another way of relating them is to use two die rolls. But now instead of a die with six faces we use N faces, where N is the total number of possible number sets we could pick in a lottery game. This “die” is really a form of Spherical polyhedron. Lets say N is 195,249,054 possible unique numbers, which correspond to each possible set. So when we roll two dice the total probability would be (1/195,249,054 X 1/195,249,054). Remember, these are “normal” die, just having a ginormous number of faces.

The above is not even mentioned in the Lottery math references, for example, this Wikipedia entry, Lottery mathematics. So some conceptual misunderstanding on my part is very likely.

Example
Lets take an actual example, the Powerball lottery states on their “Powerball – Prizes and Odds” page that to win the Grand Prize the odds are: 1 in 195,249,054. This is derived by application of math stuff to determine the combinations of the five white balls (1-59) and a red ball (1-39).

If we apply the multiplication rule the actual probability of winning is:

1 in 38,122,193,087,894,916

That’s 1 in 38 quadrillion. Big difference!
In scientific notation: 3.8122193087894916 x 1016

What is the Expected Value now?

Error?
This analysis couldn’t be correct. First, the number is too large, there are too many winners. Second, I have never heard of anything like this. Surely if this were the case it would be news. So where is the mistake?

I think it has something to do with the “same set of numbers”. Then its not just a simple multiplication of probability? If I find out, I’ll update this post.

So what?
You should not be paying the “idiot tax”. True, but when the prize reaches 100 million I bet there are some math professors out there buying a ticket too. Further, it is an interesting math subject.

Updates
This article analyzes the occurrence of a lottery draw that duplicated the same numbers and argues that my kind of instinctive analysis above is incorrect. Adventures in Probability. So, perhaps, the way to look at this issue is to compute the probability of the same winning combination being picked twice in a row? What the article says that it is 3.8 X10^16 but this has to be multiplied by the amount of combinations, so:
(1/((3.8 x 10^16) * 195,249,054))*195249054 = 1/195249054. That same as what the lottery provider quotes! I don’t get it yet. Then why are the two dice example not calculated in the same way?

Further Reading

Off Topic

Appendix
Groovy program to print the product:

x = new Long('195249054');
printf('%,d',(x * x));
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Scattered

Scattered

A short story about the future extremes of …

I remember the day it happened, about two hundred and fifty years ago. Like if it was yesterday.

My mom was getting me ready for school. She was in some ways an old fashioned mom. Instead of letting me just use the telepresenced instructors and regenerative neural training regimens, she insisted I “go” to school. So, every morning we would follow the same long routine.

The sensors would scan me, inside and out. Then the hovering drones would surround me like you see in those ancient vids about swarming flying insects. The drones would start to add hundreds of attachments, then swaddle me in a tight armor with thousands of sensors and feedback loops, monitors, and things even today I don’t understand.

Finally the last added item would be my helmet. Slowly the drones would lower it over my head, and I would always just watch my mom’s face. She had a smile, of course, but even at my age, thirty five at the time, I could see the subtle worry around her eyes, a kind of surrender. Perhaps she knew that no matter what, sending me to school was always a chance. Of what?

Mom, to keep me vigilant, was always telling me scary stories about the ancient world. Her favorite was the GooCloud demon. If you let it touch you, your life would be an endless nightmare. Your whole sensorium would be taken over, and the world would be replaced by the monster’s own dreams.

There were so many stories. Like the MicrSwamp, how it could get a grip on you and suck you down into a pit of information loss. Eventually you could not move, and life around you just kept accelerating while you kept slowing down, like falling into a black hole, an eternal trip, locked in the creatures existence.

Those monsters didn’t make sense to me; what did I know? She said that is how they grew. Something about this thing called “money”. I told her, “Your just kidding me and trying to scare me. Didn’t those things die during the Great Reset we were told about in school?” “No”, she said, “they just became bigger and more subtle”. She must know of course; she was 450 years old at the time.

The helmet snapped into place. All my breathing, sweat, everything, was tightly recycled, nothing came in or out. Even sound was processed and filtered. My eyes just saw an animated world that the suit intelligence created for me, a safe vision.

My mom said, “Privacy Protocol 108-42” and the level of activity in the fabric of our home seemed to increase. The drones surrounded me in a tight formation and the airlock at home opened. As I floated to my ‘bus’ I saw again the drones accelerate into amazing activity. My holovids were a buzz of line and shapes each trying to encircle the other. Words would flash by, “firewall 236, breach, contained; honeyed virtual proxy deployed”. And then I was whisked away toward the old school.

As I was getting off the bus, I tripped.

Even before I hit the floor, everything was changing, the feedback symbols changed colors and the sounds started to rise. When I hit, I had to shut my eyes from the overpowering wave of sound, colors, and information. The ground shook, or did everything just shake? When I opened my eyes my animated view just showed cataclysmic activity. Drones and something else out there were actually fighting. It looked like some war game from one of my alt-verse games. Fighting what?

I heard and saw a vision of my mom’s face, the smile gone. She was rapidly speaking with the intelligences in a language and symbols that appeared in the screen I could only partially understand. Suddenly one of the Ints said, “Privacy-Failure. Escalated, Life-Leak, positive; retrieve, negative. infinity pulse eminent”, and my mom gasped. All the noises stopped, the battle stopped, and I got up and stopped too. I waited. My mom’s vid just showed her staring at me, and then a single tear formed on her face and slowly, slowly flowed down.

The drones brought me back home and my suit was taken away. My mom was even in the same room, something that only happens every few years, and she crossed the floor and hugged me. I have never felt her touch before. What happened? So, she told me that when I fell my suit exposed a tiny nano-fracture and some genetic material leaked out, probably in vapor from my sweat. “Now the monsters have it”, she said. “Now they have. You.”

After the incident life went on, but then the dreams began, my sickness also. Later I learned why. Why don’t the monsters just create or reuse existing genetic information, or just our proto-animals? In ‘school’ I learned that scientists long ago determined that we humans are unique. Animals, yes, evolution, yes, but unique, nevertheless.

Our DNA acts like a tuning fork. At conception, something in the ground state of space-time, non-contextual reality, binds with human genetic structure, entangles it in science speak. The first heart beat sends a signal. To where? We don’t know. It is a one-to-one relationship. In our history we learned that clones will share in that entanglement, but imperfectly, the clones degrade and pass away, and most importantly, so will the original.

I am all over the universe. When I dream I catch glimpses of my other bodies, my other selves. I am a horde of slaves in one solar system. In another I am harvested for parts. Elsewhere I am hunted for sport and slaughtered, again and again. In a decadent world I am an army of play toys. In another world I am the experimental lab creature: burned, skinned, dismembered, gutted, and disposed. And, creatures in realms far away, use me to study humanity; to find its limits, probe our weaknesses. I am rent asunder, over and over.

All over the worlds I am dying. And, I am one of the few who is dying right here too.

Josef Betancourt
Copyright 10/4/2011
All Rights Reserved


 
 
“If you’re not paying for it, you’re the product.”
 
 


 
 
Oregon – Music Of Another Present Era – At The Hawk’s Well
Here is Glen Moore playing a piano solo that appeared in the group Oregon’s first recording. The liner notes on this track has a “(with appreciation to Meredith Monk)”

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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.