Are lottery odds incorrect, and your chances drastically worse?

September 15, 2011

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));

Eye of Odin

January 22, 2011

My Ubuntu’s (in my Virtualbox VM) background image cycles thru some great galactic images. But, now when this one comes up all I see is a left eye. The resolution and color depth heightens the affect; on my screen I even see the outline of a nose on the right.

The Helix Nebula: a Gaseous Envelope Expelled By a Dying Star
Source: Hubblesite.org

Anyway, be a good person. Your being watched! :)

The image is of: The Helix Nebula: a Gaseous Envelope Expelled By a Dying Star

Links
Hubble


On the ‘post-PC world’

October 30, 2010

What is the big deal?  If you read SciFi, everything happening now in the tech world was predicted.  Even now we are still in the “stone age” of technology.

As I read this article I’m going, “duh”?  Microsoft’s outgoing Chief Software Architect on the ‘post-PC world’ | ZDNet.  Even the web was predicted by Vannover Bush in an editorial in the 40′s I think.  The funny thing is we haven’t even implemented that, just pretty pictures and entertainment, the Semantic Web is nowhere in sight.

Wait till chip implants, robots, AI, and other things start taking off!  Thats when real change will happen; change that will cause upheavals in socio-economic balance:  what is money, what is work, what is …?


Scientist at LHC isolate the Last Minute

April 1, 2010

Large Hadron Collider (LHC), the giant expensive project to probe the inner workings of nature has already been successful.   Scientist claim that they have isolated the Last Minute.  The Last Minute has been a very elusive particle, even rarer then the Higgs Boson.  Since the dawn of intelligent humans on earth there has been an unending wait for it.  Well, now no more.  If you been putting something off, “waiting for the last minute”, its time.   Take out the garbage, empty out the closet, go for a jog, and pet your dog.

News of this discovery was greeted by awe and cheer around the world.  A housewife in Florida, USA:  “Finally, my prayers are answered! My big lazy lard of a husband will finally get off the couch and fix that leak”.

In related news, scientist from major research institutions are exploring the vast physical consequences.  They ask, “is it divisible?  Do we need an even larger collider?”   Not since the bongo drum playing of the late Feynman has the physics world been so perky.  Reports of people walking into walls are on the increase.  String theorist are itching to write even more inscrutable papers but their fingers are tied up in knots.  Philosophers are falling into comas when recursively contemplating what happens to time when Last Minutes are used up.

Is the Last Straw next?


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