Statistical Question (2000 SR1)

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Assuming a population which is entirely random numbers, which of the following assumptions would be “more” correct?

1. That the most frequently occurring numbers would occur next, or…

2. The least frequently occurring numbers would occur next since they have not occurred very often.

Thanks,

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Neither. If the numbers are random, then every possible number has the same probability of comming up next, no matter what has happened before.

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Even if the population is finite?

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I agree with Legare – any number.
I think it is a matter of separating Possibility and Probability.

Take flipping a coin. You have two Possibilities – heads and tails.
If you fip and get heads, the Probability is the next flip will produce tails as the overall odds are 50:50 and statistically you can expect an even spread.
However, the Possibility is that it will be either, as each flip has an equal chance of producing a head or a tail.

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If the selection process is random, it does not matter if the population is finite. On any selection, the population is always finite – the current population plus one.

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If the sample is finite, then there may well be a difference.

For instaance, if, in a pack of cards, 12 hearts have been drawn (12 cards out of 52, leaving 40 cards), then the chance of the next card drawn being a heart is 1 out of 40 (vs 1 out of 4 of the first card drawn being a heart).

But the population must be very finite for this to happen.

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Hi:
When sampling, without replacement, from a finite population, that originally had an equal distribution of elements at each value, then your second premise would be true: “the least frequently occurring numbers would ” have the the higher probability of showing up on the next draw.occur next since they have not occurred very often.

was formed by a random process such that all values had an equal chance of being included to the same degree, then

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After reading Geoff’s response, I think I may have misunderstood this reply. If by finite population, you are refering to the population from which the numbers are chosen, then the results are not truly random, and I agree with Geoff’s response.

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Legare,

I think there’s a misconception out there- common amongst gamblers.

“If I’ve thrown 20 heads in a row- what’s the chance of heads being the next throw”

“Mr Jones has had 6 children- all boys. What’s the chance of the next child being a boy?”

“This poker machine has not paid for the last xx. So it must be due for a payout now”.

I’m not sure if that is behind the original question.

The probability of any random event occuring is completely independent of any previously occurring random even. As long as both events truly are random.

We’re getting way out of scope of an Excel forum. I guess that Excel just attracts the interest of statisticians and gamblers

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I typically see this kind of question from someone trying to create a spreadsheet that will predict lotto numbers given a list of previous numbers.

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U got the prize! Actually I was trying to dislodge a conviction by a coworker that the numbers which are drawn most frequently are more likely to be drawn next. She has some interesting data, but I think to be mostly coincidental.

Thanks,

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Then, there are some things that you can say about the probability of a number comming up:

1- There is no change in the probability of a number comming up based on numbres from previous drawings. If the number 1234 has hit every day for the last week, it still has the same probability of hitting today that it did one week ago.

2- Depending on how the numbers are drawn, there may be a difference in the probability of certain numbers in the set of all possible numbers. For example, if the ping pong ball machine is used where each digit drawn is not replaced before the next digit is drawn, then the probability 4444 being drawn is less than the probability of 1444 which is less than 1244 which is less than 1234.

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Something I think might produce a more meaningful result, is “What numbers pay out the most?”

A typical lotto game might have 40 -45 numbers. But the numbers icked are probably not entirely random. Many people would have special numbers based on birthdays, anniversaries, “lucky” 3,7,13 or the like.
If that is indeed the case, then numbers above 31 are picked less often, and when they do win a prize, are likely to get a bigger share of the pool. If that is the case, then you don’t increase your chances of winning by using those numbers, but you would get a bigger payout if you did win.

I hope your cow orker gets lucky.

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Actually, Geoff, I think I have put the question to rest. I told her that in order to draw any sort of forecast, prediction, or other conclusion outside of which numbers occur more frequently and which occur less frequently, the population would have to be related in some fashion. In other words, a set of 5 or 6 numbers drawn in one drawing would be related in that any single number could not be drawn twice. However, when looking at multiple drawings, there is no relationship between the numbers drawn other than the fact that there is a min and a max.

I even graphed the results of the last 30 drawings, 20 in one set, and 10 in the other. The set with 10 drawings had little relationship to the set with 20 drawings.

For what it’s worth, I think no matter what I say, she still is convinced that her system will eventually pay. I guess if she hits, I will be eating crow…

Thanks,

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Mike,

She’s happy. But I would like to know myself if humans do select numbers randomly or non-randomly. I’d suspect the latter, but I say that without anything to back it up. Except that I’m a human .

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I suspect that you are correct. Hans once stated that “If you torture numbers long enough they will tell you anything you want.” When you look at a long list of seemingly unrelated numbers, such as a drawing, and you see 8 selected 27 times, and 34 selected 2 times, you have to wonder HOW? Supposedly the drawing is random, but I would guess that looking at end results and inadvertently extrapolating a cause, or forming a conclusion regarding the nature of future events based on past events IS human nature!

For me, Friday the 13th went perfectly well!

Thanks,

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A clarification would help — once a number is selected from the population, does the size of the population left to be selected from decrease? Or, does the number get tossed back into the hat possibly to be drawn again?

If a selected number is not put back, its frequency within the pool will be decreased, increasing the probability that any other number will be selected next. If the number is returned to the pool, it has the potential for being drawn again, and none of your initial probabilities will have changed.

Even if selected completely at random, a number that occurs more frequently within your population will always have a greater chance of being selected. A rare number may never be selected, or it may always be selected; a common number may never be selected, or it may always be selected. Probability dictates that a rare number is more likely not to be selected, and a common number will be.

Hmmm… I think my stats prof would be proud — something sunk in!

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[indent]

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something sunk in

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[/indent]Do you really sink so?

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Thanks for everyone’s input.

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If you know the frequency of the presence of the numbers in the finite sample, e.g. you have a sample of 100 numbers and you know that there are 10 numbers which equal “5.5”, then you can use the binomial function in Excel to calculate the probability of finding a “5.5” when selecting only 1 number out of the 100 numbers in the sample:
=BINOMDIST(1, 100, 0.1, false).
Of course this will be the same for every number with the same frequency. This allows you to do some simulations, changing the frequency of occurrence (= the prior probability). This only applies if the samples are taken out independently and when they are replaced. It is clear that if they are not replaced, the frequency will decrease, but also the sample size and this should be taken into account. e.g. suppose you took out a number “5.5”, then there are only 9 numbers “5.5” left in a sample of size 99, then the next attempt has a chance of =BINOMDIST(1, 99, 0.09, false) to be succesful (meaning to be equal to “5.5”)

You see, we were not drifting away that far from Excel as all calculations can be done with built-in functions.

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