# Thread: Help understanding rarity vs. probability…

1. ## Help understanding rarity vs. probability…

I am having a little problem understanding the difference (If there is one.) between an event being ‘Rare’ versus the probability of that event happing.

For example;

The English lottery has 6 numbers (ignoring the bonus ball for now.) If I take the average of all possible combinations of 6 numbers it forms a normal distribution, with the mean of the population being 25 and the standard deviation of about 5. (5.4645)

I deduce from this that if I take a random sample (i.e. pick any of the previous week’s lottery results at random.) and take the average of the 6 numbers, 95% of the time the average should be between 15 and 35.

From the above information, the numbers {1,2,3,4,5,6} who’s average is 3.5, would be a very rare occurrence. Yet I know that they have the same probability of being drawn as any other 6 numbers.

So my question is;

If I have a process at work for example, that has a mean of say 50, a standard deviation of 0.5 and it is normally distributed, if I were take 1 random sample out of that process I am guessing that it would be a rare occurrence for that sample to measure 51.5

Assuming that the various factors that control the output of the process have an equal chance of happening, would that 1 sample that measured 51.5 have the same probability of being picked out as any other sample? Even though I know that 95% of the time the sample was likely to be between 49 and 51.

p.s.
Just out of curiosity, if everyone who played the lottery picked numbers whose average was between 15 and 35; would there be a lot more winners??!!

2. Your six numbers do NOT come from a Normal Distribution. It is a uniform distribution. The Mean of only 6 picks isn't Normal, either. After that grave misidentification, everything else you calculated or wondered was simply wrong. No wonder you found it confusing.

In the absence of cheating, lottery numbers are independent. Each has NOTHING to do with the previous. Don't waste any money trying to leverage anything.

Your six number do NOT come from a Normal Distribution
Yes I know this. But it's only the means of any 6 numbers that I am looking at and not the probability of picking 6 correct numbers.

The Mean of only 6 picks isn't normal

I am not sure what you mean by this. If I took the means of a number of samples of 6 numbers or indeed the means of each possible combination of 6 numbers (using numbers 1 to 49) they do form a normal distribution. I am not too sure but isn't that a way to normalise data? Take the means of samples? (Let me know if I’ve got that bit wrong please.)

So as far as I can see my question is still valid. However I do see that maybe I am not comparing like with like. The probability of picking any 6 numbers is different than the probability of picking 6 numbers whose mean is a certain value.

In any case I was only using the lottery as an analogy for understanding processes and I think I’ve unwittingly answered my own question.

Cheers
Emerson

4. Originally Posted by emersong
I am having a little problem understanding the difference (If there is one.) between an event being ‘Rare’ versus the probability of that event happing.

For example;

The English lottery has 6 numbers (ignoring the bonus ball for now.) If I take the average of all possible combinations of 6 numbers it forms a normal distribution, with the mean of the population being 25 and the standard deviation of about 5. (5.4645)

I deduce from this that if I take a random sample (i.e. pick any of the previous week’s lottery results at random.) and take the average of the 6 numbers, 95% of the time the average should be between 15 and 35.

From the above information, the numbers {1,2,3,4,5,6} who’s average is 3.5, would be a very rare occurrence. Yet I know that they have the same probability of being drawn as any other 6 numbers.

So my question is;

If I have a process at work for example, that has a mean of say 50, a standard deviation of 0.5 and it is normally distributed, if I were take 1 random sample out of that process I am guessing that it would be a rare occurrence for that sample to measure 51.5

Assuming that the various factors that control the output of the process have an equal chance of happening, would that 1 sample that measured 51.5 have the same probability of being picked out as any other sample? Even though I know that 95% of the time the sample was likely to be between 49 and 51.

p.s.
Just out of curiosity, if everyone who played the lottery picked numbers whose average was between 15 and 35; would there be a lot more winners??!!
Forget the word rare, it does not have a well defined meaning in the context
of probability theory. Stick with the probability of events.

RonL

5. 1) Your six draws are not quite independent, usually being from a small fixed number WITHOUT replacement.

2) Normal is a very poor approximation for MultiNomial distributions for very small sample sizes.