The Law of Averages

 

A friend of ours has managed to simplify statistics and probability and distil it to its very essence. This man’s philosophy is that either an event happens, any conceivable event, or it does not. This man is a computer programmer of all things.

 

For instance say you drop a slice of bread, the chances of it landing butter side up is fifty-fifty. Or you wake up and no-one’s around and no one answers a phone then the chances you missed the rapture is fifty-fifty. Of course the Lotto is a nice one? Play just one ticket your chances are half-half.

 

Fifty-fifty is the probability. Always. There is then nothing unexpected and no surprises, ever. To me it sounds like a very sound approach even without contradiction- logically as well as practically.

 

 

To illustrate the idea we use as an example repeatedly casting dice. The chances of throwing say a three is p(3)=1/6 assuming then a die is equally likely to land on any one of its six sides. Of course the chance to throw any e.g. a one, or a six, etc. is the same, each is 1/6 out of symmetry.

 

We are looking then at the fraction of the number of three’s divided by the total number of throws, or experiments. The idea is then that after “very many” throws you will find this ratio will get closer and closer to (approaches) one sixth, the same as its probability.

 

The “Law of Large Numbers” is an exact mathematical formulation of what is commonly known as the “Law of Averages” which concerns repeatedly performing the same experiment, indefinitely, each time with the same fixed probability p of success. In everyday language, the law implies that the ratio of successes to total number of trials quickly nears and eventually evens out to its probability p.

 

This theorem has many refinements and can of course be generalised and sophisticated in many ways.

 

Probabilities can in principle also be defined the other way round. You could as easily do experiments in such a way, and estimate a probability value from the results. For instance it could be that you don’t have any direct way of calculating theoretical values.

 

Scientists and gamblers believe the law has been proved as a mathematical truth but mathematicians seem to think the law of averages is a law of nature. There is a fundamental difference. Personally, the most amazing thing of statistics and the theory of probability is that it actually works. Apparently!

 

‘Torture numbers, and they will confess to anything’ –Gregg Easterbrook

 

 

What does it mean, “there is a 60% chance of rain tomorrow”? Is there a difference between “scattered thunder showers in the afternoon” and “a light drizzle in the morning”? Do both count as “rain”? And this as a number, 60% , what does this mean? Where does it come from? Look, it either rains or it doesn’t and in either case I maintain the number in itself is just about meaningless.

 

‘He uses statistics as a drunken man uses lamp posts – for support rather than illumination’

–Andrew Lang

 

Our statistician friend had an average of only one pint of beer in the evening. He drowned in the gutter one friday night of alcohol poisoning.

 

‘The average person in Cincinnati has sex 1.45 times a week’. Somewhere someone is doing something wrong!

 

 

I heard this story from an old man you might know, it was of a certain Jewish gentleman who came to visit his friends and family overseas. He was taken on an outing to the horse races which he thoroughly enjoyed, and afterwards he was asked what he thinks and does he want to play? He said he’d loved to, it’s wonderful and yes he wants to play but he wants to be on the other side of the counter.

 

The bookie is guaranteed to win as a whole with each race irrespective of which horses win. The odds are numbers so calculated, to offset the bets. Bookies don’t gamble.

 

 

Fail-safe indeed. If you read your insurance policies carefully, please take note that there is no cover at all in the case of natural disaster, civil war, war, and all other different kinds of large scale catastrophe.

 

Insurance “cover”? The only thing an insurance company ever really covers, is itself.

 

Yes and of course the kind of soul insurance often sold on TV is absolutely fail-safe since there are never come-backs. Big church is big business.

 

 

I hope to still provide some sketches to explain the ideas better.

 

 

Many people spend a great deal of time in pursuit of a guaranteed “system” in gambling games such as roulette. Recently I was told of this one.

 

Put one chip on red, if the ball falls on black, put down two chips on red, suppose that the ball falls on black again then put four on red. The idea is that once (inevitably) it falls on red you have won and “doubled your money”.

 

Not true. If you keep track of the amount you've already lost! Suppose you win on the fifth spin. That means you've placed 1 x 2 x 2 x 2 x 2 = 16 chips on red, and that is what you've “won”. However you must keep track of what you've already lost! This is then 1 + 2 + 4 + 8 = 15 so in total you sit again with only a single chip! Which is what you started with! The idea can be generalised for any amount of spins by using the high-school formulas for geometric progression.

 

The big problem here is that your stakes increase exponentially, astronomically. To keep it up, by the eleventh spin you already have to put down 1024 chips and for the 21st spin you must have well more than a million chips! It could be that you reach a point quickly where your funds dry up and you can't play any more in which case you've lost all of it!

 

In the long run like this you do break even but it’s not a good idea, it is highly impractical it could even be very risky, as well as being very monotonous.

 

The method obviously doesn't work. The other problem inherent of Roulette is of course the zero, or black, always giving the bank a slight advantage. To me it seems that all you're ever guaranteed of is that that eventually you lose your money, the only difference is how rapidly.

 

The casino always wins. No strategy will work.

 

 

For interest’s sake. The question arises whether your chances at winning are better if you play more tickets. The common belief is that if you play two tickets, your chances double. The surprising fact, the good news is your chances of getting either on ticket is incrementally (in effect infinitesimally) more than double. The catch is in replacement.

 

One can see it like this: Let us put ten marbles in a circle and you choose a marble. For someone then picking one marble at random the chances of it being yours is 1 out of 10 or ten percent.

 

Suppose now instead you had chosen two marbles to start with. For the first marble the probability is still 10% but for the next one, the second, it is now 1 out of 9 since only nine marbles are left to pick from. Which makes your chances of getting the right marble altogether 10% + 1/9 = 20.111.. % and is slightly more than double.

 

The catch is in replacement. However as you increase the initial total number of marbles (tickets) the effect rapidly diminishes and becomes completely negligible.

 

If you had put back the first marble and picked at random again, then your chances as a result will indeed be 20% but it does sound stupid to choose the same ticket twice, repetition, unless of course you play two games, or two different Lotto draws! Makes sense hey? All your eggs in one basket, a single draw! Double or quits, one shot.

 

Ok in principle, do my chances double each time I chose an extra ticket? Certainly not. It doesn’t work that way. One can for instance repeat my example calculation but choosing three marbles. If you choose three numbers your overall chances will only be slightly more than three times.

 

We must realise this all says almost nothing. You should actually be interested in your expected winnings, or expected return. Which in fact is a small (and negative!) amount. For instance if you buy more and more tickets your odds of having to share “the Lotto" prize becomes larger and larger making your expected return in winnings all the smaller.

 

 

In repeated sampling and measurements of physical data of the same quantity the normal distribution seems to occur almost universally in nature, as so it does in mathematics. Given that individual measurements can be viewed as independent and identically distributed, it is found that the spread of recorded values corresponds to the normal distribution.

 

To illustrate let us look as an example at yearly rainfall figures. Make a table of total rainfall in each year since 1914 and record it as millimetres per year and graph the frequencies. The peak will be at the average and diminishes as the discrepancy increases. You might say in a year of drought there would be 10mm's of rain the whole year and let's say we had two years of drought in the century. On the other hand if there where terrific floods, say the rainfall in such year was 200mm's there would have also been only two three instances in the century.

 

The mean of the graph would then be at the average yearly rainfall over the century, and it would be much more likely for the majority of years that the measured be closer to the average.

 

So you plot measured rainfall frequencies. Increasing the sample size (amount of years) makes the graph all the smoother and more reliable. Please note that there is no guessing here!

 

The normal distribution curve will look as follows. The mean (average rainfall over time) is in the middle where the graph peaks. The curve is symmetrical. The area under the graph must equal one, related to a probability of One for an event that has to occur. As one deviates from the average the frequency decreases dies out and the curve flattens to zero. Depending on the phenomena and kind of data the curves are usually scaled (transformed) to give your typical “road-bump” shape kind of curve.

 

As a limiting case the normal distribution is the familiar bell-shaped curve. The graph is characterized by two parameters. The mean, or average which is the maximum of the graph and about which the graph is symmetric, and the standard deviation which determines the amount of dispersion away from the mean and the shape of the graph.

 

This was an example in nature, there are many, and many quite concrete examples in probability theory also. In fact for these kind of investigations in mathematics the original motivation came from gambling games.

 

The Central Limit Theorem establishes the normal distribution as the distribution to which the average of almost any set of independent and randomly generated values rapidly converges. It explains why the normal distribution is such an excellent approximation for the spread of a collection of data.

 

‘Everyone believes in the normal law, the experimenters because they imagine that it is a mathematical theorem, and the mathematicians because they think it’s an experimental fact.’

–Gabriel Lippman

 

 

Flipping a coin is in principle the exact same situation as Schrödinger's famous cat-in-a-box example but mine is a counter-argument.

 

Think of a cup turned upside down over a coin. The coin is heads or it is tails. It already is. The act of lifting the cup has no effect on the outcome. Indeed the act in itself does not result in the coin being heads, or tails. The difficulty is information. It has in fact already been determined, but you don’t know which, you have no knowledge.

 

So to just flip a coin I maintain, is the exact situation. It will have fallen on heads or tails. It is a fact. Either. But you do not have means of knowing before-hand which. It is a matter of prior and present knowledge. I believe this universe evolves deterministically and a thing such as “chance” does not exist. Probability Theory is merely a method, a way for us to deal with information and phenomena we do not fully understand.

 

Even said, in all, statistics probably works a lot better than fuzzy logic. I guess? Not?