No, I am not making this claim. I'm making a different claim.

I am saying that the entire set of possible game outcomes can be enumerated, thus giving a 1:1 correspondence with a set of integers between 1 and N (where N is the number of possible outcomes).

Once you do this, the ENTIRE SET of drawn balls can be equated with an integer. Each set is equated with a unique integer.

This makes the lottery draw (the ENTIRE SET) equivalent to the random number generator which returns an integer between 1 and N.

I don't see what makes this so hard to realize.

Back to the question at hand:
Offer this to your friend: "I'll put a dollar in an envelope for every lottery ticket you buy. At the end of the year we'll compare your winnings (or lottery stubs) with the dollars in my envelope."

Could be interesting.

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The concept is less difficult than writing clear explanations.

First off, can we agree that time is not a factor?

What I mean is that the order in which the samples (balls) are pulled wont affect the probability of the final combination of balls.

Secondly, can we agree the the lottery system we're fussing over is sampled without replacement?

If both of these characteristics are true than what we have here is Matt L describing a problem in combinations while Chas H and I are describing the SAME problem in terms of probability.

From the perspective of Matt L the probability should derive from the total number of non-repeating combination possible.

k_C_n = n!/k!(n-k)!

Where n = the sample set (or population size) and k = non-repeating combinations. What is this?! Here's lottery sequence: 55 21 39 76. In terms of possible combinations it is the same as 76 39 21 55 but not the same as 55 12 93 67, etc. Also x! Continuing product = (x-0)*(x-1)*(x-2)...(x-x-1).

So if the balls are numbered 1 to 100 and are drawn as a subset of 4, the possible number of non-repeating combinations = 100!/4!(100-4)!

I'll bet that can be converted to a probability, but I'll have to search the WWW for how it's done.

I think you're getting it, Bot.

We're not betting on each number to come up, knowing the prior results. If we were, the fact that we sample with or without replacement would matter.

The fact that we sample without replacement simply reduces the number of possible outcomes. The fact that the order doesn't matter reduces it further. But there is still one list of all possible outcomes (modulo order).

I'm not talking about a random number generator which returns a number between 1 and 55, and is run six times.

I'm talking about a random number generator which returns a number between 1 and 130,000,000 (or whatever the number of combinations is). This generator is run a single time to represent all six balls. You take the returned integer, and look up the combination in the pre-prepared list. Since the list won't contain any duplicates, either the same ball present twice or the same set being present twice in the list, this is an equivalent activity.

I think your method is a solution to the method of computing the combinations by calculating the permutations checked against a look-up table to eliminate duplicates, deriving the combinations. Good solution and I'll bet you could program that in C easier than writing all the damned "Do Loops" necessary to derive a number from the formula.

But that does NOT produce a probability but merely a list of possible outcomes. A combination is not a probability.

This is true, although the table would be quite large. It would be better to write a function giving the mapping, if possible.
A complete, finite list of equiprobable outcomes implies the probability of each. In terms of the above, each possible pick has a probability of 1/N.

Now you know enough to compute a probability both ways.
Does PI 1/(x-i) = Matt L's method = n!/k!(n-k)! ???

Go forth and multiply!

B

Your speaking gibberish jargon, that why it's hard to realize. Post up some examples as I did to illustrate your point.

It may involve some jargon, but it's anything but gibberish.

Here's an example for you. It is necessary to make the game much smaller in order to list the possibilities.

Say that you have an urn of five balls, and you choose four, without replacement and disregarding order. That is, exactly how the Powerball works, but with a much smaller game.

Here is the list of possible outcomes, with an ordering:

1: 1 2 3 4
2: 1 2 3 5
3: 1 2 4 5
4: 1 3 4 5
5: 2 3 4 5

I told you that it would be a small list. I leave it as an exercise that this is a complete list, but it is. But the idea works for any number of balls in the urn, and any number of drawn balls.

Now, instead of actually picking four balls from the urn, you roll a fair five-sided die and look up the outcome on the list.

So to replace the bin holding 50 balls we roll a die with 15,625 x 10 to the 6th power sides? That's going to be a damn big die.

Indeed, it's a huge die. But instead of a die, you use a fair random number generator, with the appropriate range. Of course, computers use pseudo-random number generators, and need not apply.

Hopefully you can see the pattern used above, and how it extends to any number of balls.

For a draw of six balls out of 55, you have a sequence like this:

1 2 3 4 5 6
1 2 3 4 5 7
...
1 2 3 4 5 55
1 2 3 4 6 7
1 2 3 4 6 8
...
1 2 3 4 6 55
1 2 3 4 7 8
...
...
49 50 51 52 53 54
49 50 51 52 53 55
50 51 52 53 54 55

Just fill in those monstrous gaps, and you have a list of more than a hundred million entries. But it's still a list, and thus a mapping of a finite subset of the integers to all possible (and equiprobable) lottery draw combinations.

Will we see this on the Powerball drawing anytime soon?

If you fill in the entries in that list (and ignoring the separately-drawn extra ball), you will see one of them on the very next Powerball drawing. Which one you see is completely random, and the chance of each is 1/N, where N is the length of the list.

Will we see your method of choosing the winning number used by Powerball?

Of course not. What I'm saying is that when the numbers are drawn, it is equivalent to choosing an integer from 1 to N (where N is somewhere around 130,000,000, if I recall correctly; the actual number matters little). Just take the list of balls, and look it up on that huge list, and you have an integer.

The Powerball is just a random generator. Nothing more, although it is truly random. If you accept that, you should also accept that any random generator which outputs the same range is an equivalent action.