Tag Archives: probability

Quora Answers 8/7/2015: Random Events and Causality

Why do you believe that random things do not follow cause and effect?

Let us think about this a little more, shall we?

What do you think it means for something to be random?

Let us take the roll of a die. What do we mean when we say that the outcome is random?

Do we mean that the die does not follow cause and effect? No, of course not. At least, not if we are being rational.

We know that if we roll the die, that cause and effect is in play. The die does not move around according to some magical forces. It moves around due to a complicated chain of interactions with the air, the surface it is being rolled across and so forth.

We do not say that the die lands on, say, six, for no reason. No, there is a cause for it showing a six. But the series of events that caused that to happen is a complicated one and we have no way to predict the outcome.

Or, let us suppose that we are an insurer. We have no way to predict if a particular client is going to have a car accident. We acknowledge that if a customer does experience a car crash, there is a reason. There is some chain of cause and effect.

We know that the car crash has a cause. There was one or more event that logically led to the car crash.

Maybe the driver was distracted and he did not see the car coming towards him. Or maybe his brakes failed or whatever. But there is some causal link between one or more event and the car crash.

As an insurer, we do not know what will cause these car crashes in advance. But we can estimate how often on average our customers may crash their car. We may figure that, say, 1/100 customers will experience a car crash.

If we know a bit more about particular crashes, such as their past driving history, we may be able to estimate that that particular customer may have a 5/100 chance of having a car crash at some point.

What does this have to do with random events? Notice that when rolling the die or trying to estimate how often people crash cars, we are trying to estimate how often to expect certain results.

dice, random
Unpredictable little things, but they are not magic..

We have situations where it is hard to predict the results, but we do have mathematical methods of estimating how common certain outcomes might be.

When we say that something is random, we are saying that “We are unable to predict with certainty what outcome to expect. But we know it will be one of these known possible outcomes”.

In other words, randomness is an epistemological issue. Randomness simply indicates that are unable to be certain what the results are and can only guess what the outcomes might be and possibly how frequent certain outcomes might be.

But our inability to predict outcomes does not mean that there is no causality. Just because we cannot predict outcomes does not mean causality does not apply.

Something is random when we have no means of predicting the outcome with any certainty.

Now, in the statistical sense, something is random when we cannot predict out outcomes but we know that every possible outcome has an equal chance of occurring.

But we know that something is going to happen. But how does it happen? Is there any cause and effect? Yes. Just because we cannot predict what will happen does not mean that cause and effect does not apply.

Whatever outcome does occur is because something happens and the nature of the relevant entities means that that outcome had to happen. That is, cause and effect.

That is what causality refers to. That if these entities do this, then the natures of the entities mean that this other thing must happen. There is no alternative, the nature of the entities involved requires that outcome. There was no alternative.

Our inability to know what the outcome is in advance is not an argument against cause and effect.

This applies in the quantum world. There are no truly random events. Everything that happens in the quantum world is because that outcome is what had to happen when the quantum entities do whatever they are doing, there was no alternative.

Take two particles that interact and particle A causes particle B to fire off at that angle. That is what had to happen due to the nature of the two particles.

There is no sense in which there was another possible result.

What about the fact that in quantum mechanics things can lack definite properties?

That has never been established and never will be. To exist is to exist as an entity with a specific nature, there is no alternative. Nothing that exists has an undetermined nature or a contradictory nature.

Therefore this does not provide a rational objection to causality. Everything that exists has a specific nature and will do a certain thing under certain conditions. What it will do will depend on its specific nature.

There is no way around this. Everything will do what its nature requires it do and nothing else.

The fact that we cannot always predict what it will do is not an argument that it will not do that thing. It just means we can only guess what it will do and we should try to use probability to predict how often certain outcomes might occur.

You can find out more about that here.

You can check out the Quora question and some of the other answers here.

Statistical Fallacy List, Part Two

If you have worked with data, then I bet you have been guilty of one or more of some kind of statistical fallacy at some point. I know I have!

In this series, we will be looking at fallacies that often come up when analysing data or, allegedly, academic sources.

This is part two. You can find part one here.

Simpson’s Paradox

A paradox? How is this a fallacy? I thought that we were talking about fallacies?

We are, but we must introduce this paradox first.

Simpson’s Paradox is a phenomenon where a trend appears in different groups of data but vanishes or reverses when those groups are combined.

The classic example of this is a study performed in Berkley University in the 1970s over the following data.

Note that the last row shows the total application success rates for both genders. If you look at this data alone, it might seem to suggest that the application success rate for men is higher than that of women.

This led to Berkeley being accused of sexism. But is it as simple as this suggests?

If we look at the data, we notice that 1,800 women applied for subject A. Only 168 men applied for that same subject.

Of those 1,800 women that applied for subject A, 15% of them were approved. While 14% of the men that applied for subject A were approved.

This is a slightly better result for women. When it comes to applications for subject A it seems that Berkeley was not being sexist.

Let us look at subject B now. For this subject, 50% of men that applied were approved and 51% of women were approved. Again, this is a slightly better result for women and it is hard to argue that Berkeley was sexist.

How then, to explain the fact that women seemed to have a lower overall success rate?

Let us consider that subject A seems to have a lower approval rate for both genders. It seems Subject A is a competitive subject with very low approval rates for both genders.

Now, out of 2,000 applications for women, 1800 of those were to subject A but only 270 of these were approved. That is, 13% of these applications were applications for subject A.

For men, out of 2,000 applications for men, 1200 of these were to subject A but only 168 of those were approved. That is, 8.4% of these applications were applications for subject A.

Note that 1800 out of 2,000 applications made by women were for subject A. That is 90% of applications. Whereas for men, only 60% of those applications were for subject A.

A significantly higher proportion of women were applying for subject A, the subject with a much lower approval rating. And most of those applications were going to be rejected.

So, it stands to reason that a higher number of women would have their applications rejected.

math
Maybe subject A was a hard class like advanced math of physics??

So, far from Berkeley being sexist, the real reason women had a lower application success rate is that they tended to make more applications to subjects with a lower application success rate.

Let us see what happened here. If we look at the data for subject A and B, we see that men and women have about the same chance of being rejected for each subject. With subject A having a much higher chance of rejection for both genders.

But if you combine the subject rejection rates for the genders, you get a 28% rejection rating for men and 19% for women.

This seems to suggest that women are rejected more often than men, even before you did this, this trend did not show itself.

This is Simpson’s Paradox at play. It is the statistical phenomenon where trends disappear or reverse when you combine data.

In this case, women had a slight advantage in application rates if you do not account for the proportions of women applying for subject A and add the acceptance rates per subject together for each gender.

If you do this, the trend reverses and women have a lower application success rate!

The fallacy would be failing to recognize why the trend seems to reverse and assigning some erroneous cause.

The reason we observe the lower application success rate for women is not sexism, but the fact that a higher proportion of women are applying for a more difficult subject.

When you see Simpson’s Paradox you should study the data and try to identify the cause for this paradoxical disappearance or reversal of trends. Not simply assume some erroneous cause.

Try to avoid the fallacy of misinterpreting trends in the data.

headache
Yes, it can be hard to see the patterns in data.

Let us take one more example before we move on.

Suppose we have two baseball players, Joe and Martin. During the years 2019 and 2020 we have the following data:

Note that in both 2019 and 2020, Martin had higher batting averages. However, when you combine these years, Joe has a higher batting average.

What gives? This is caused by the fact that Joe had a lower batting average for both of these years but a lot more time at-bat, meaning that when you combine the data he has a slightly higher batting average.

You could assume that the data was rigged or that maybe Martin was a better batter after all. But that would be a fallacy.

The real cause of the fact that the combined totals being better for Joe is more to do with the fact he spent more time at-bat.

If you see trends vanish or reverse when data is combined, always look more carefully at the data and see why this might be the case. You are likely to find that there is a perfectly logical reason this happens that has more to do with the data than anything else you might erroneously assume.

Statistical Fallacy List, Part One

If you have worked with data, then I bet you have been guilty of one or more of some kind of statistical fallacy at some point. I know I have!

In this series, we will be looking at fallacies that often come up when analyzing data or, allegedly, academic sources.

Biased Sample

This fallacy arises when you do not take a representative sample from a population.

What does this mean? What is a sample and what do I mean by a population?

In statistics, a population is a set of all the things one is gathering statistics on. It is the collection of all the things you are interested in studying and getting data on.

For instance, if you are getting statistics on the height of males in the US, then the population is all the males in the US. If you are studying the lifespan of fruit flies, then your population is all fruit flies.

A sample is a subset of a population that is chosen as representative of the population in general.

Usually one cannot get data on the entire population. One is not able to measure the height of every male in the US or the lifespan of every fruit fly in existence.

If one is doing a poll on political views, then one is unlikely to be able to ask everyone in the population what their political views are.

So, one must take a sample of the population. They have to select a subset that is assumed to be an accurate representation of the population.

So, if one is interested in the height of men in the US, one picks a bunch of men and infers things about the height of men in the US from this subset of men.

Or, if you are interested in studying political views, you pick a sample of people in the population and ask them about their views.

The sample must be a fairly accurate representation of the population. The sample must be chosen so that it is valid to analyze the sample and use information about the sample to form conclusions about the population as a whole.

The subject of sample selection is complex and we will not go into it here.

Suffice to say that a proper sample must be taken so that the sample is sufficiently representative of the population.

What this fallacy deals with is the situation when the chosen sample is biased and does not accurately represent the population.

This often happens intentionally when people choose a sample so that it seems to prove their assertions about the population.

For instance, suppose I want to show that Scientology is a growing religion. However, I mostly survey people with known associations with Scientology. This creates a bias in my results that does not accurately represent the population as a whole.

Let us suppose I wanted to do a study on global religious views. But I took my sample mostly from the Vatican. Yeah, my sample might be more than a little biased…

Suppose that I want to sample the height of men in the US. Then I probably do not want to sample only men that are over 7 feet tall. This will not give me an accurate picture of the average height of men in the US!

If I want to get an idea of the attitude towards Communism in the US, then I probably do not want to sample only Communists or only those opposed to Communism!

In other words, I do not want to choose my sample so that it misrepresents conclusions about the population.

The problem is that I run the risk of results that are not representative of the population. My results indicate trends that are a result of the way I selected my sample and are not truly indicative of the population.

I need to select my samples to accurately represent the population and not cherry-pick a sample that seems to make the point I want to make.

Gamblers’ Fallacy

This is named after the fallacy typically held by gamblers. As well as many other people engaging in games of chance and the like.

Suppose that you are betting on the roll of two six-sided dice. You notice that the dealer has rolled 10 a lot in the last few rounds. You, therefore, assume that he is less likely to roll a ten the next time he rolls the dice.

This is however not the case. For statistically independent events, it does not matter what happens in that past, any outcome always has the same probability of occurring.

Events are statistically independent when every possible event has a certain probability that is not affected by what has happened before.

That is to say, the outcome is not affected by previous outcomes.

Dice do not care what you rolled before, the chances of rolling a given number are exactly the same every time.

Therefore, it does not matter if you roll ten on a dice ten times in a row. The chances of rolling ten on two six-sided dice are always 1/12, even if you just rolled ten one hundred times in a row.

Past events, good or bad do not affect the odds of statistically independent events.

A typical example would be when you assume that because you have had a streak of bad luck, that you are due for some good luck. Say you play Lotto and you fail to win anything for ten years but assume that after all this time that you are bound to win something one day soon!

This is not the case; you are no more likely to win Lotto now than ever before.

Or suppose you believe that since you had three girls in a row, this time you will most likely get a boy. No, you are just as likely to get a fourth girl as you are your first boy. The odds of getting a boy or girl are still 50/50.

Or you assume that because you have been rejected for five jobs in a row that today you are more likely to get one. No, you are just as likely to get it as you were as if the five rejections had never happened. All else being the same of course. And assuming you leave everything to dumb luck instead of improving your chances by upskilling.

Streaks of good or bad luck are meaningless and do not affect the outcomes of future events.

probability, dice, die

The Role of Probability in Science.

Today we are looking at the proper role of probability in science. And how this has been subverted by modern physics. Let us first look at what probability is.

Let us suppose that we want to roll a six on a six-sided playing die. We know that on average, we will roll a six about one time per six rolls of the die. That is, if we roll the die six times, we would expect to get a six about one time.

We could throw the die twenty times and get six zero times. But we would be surprised if this happens. We would consider this to be a very unlikely event.

The more superstitious person might consider oneself cursed!

We say that the probability, or chance, of rolling a six is one in six. But what do we mean by this? What does the concept of “probability” describe?

There is much that can be said about calculating and determining probabilities in a huge range of imaginable contexts. Mathematics has much to say on this topic and we can use it to calculate/estimate all sorts of probabilities using very sophisticated methods.

But we will keep things simple and ask what the concept tells us about reality, using simple examples.

Probability is a method of dealing with uncertainty.

It takes certain processes, such as rolling a die or predicting the weather. These kinds of processes are very difficult to predict. We have little ability to be certain what the outcome(s) of these processes might be.

For instance, consider rolling a die. Although it seems to be a very simple process, we cannot track all of the tiny motions of the die. We do not have any way of knowing exactly how it will dance through the air and then strike the table.

probability, dice, die

Such simple looking things, but it is next to impossible to predict the outcome of throwing them!

Or consider the weather. This is an extremely complex thing to predict. Countless factors go into determining the weather on any particular day or even hour. So many that we currently have no reliable way to account for them all and we likely never will.

What do we do? If we cannot account for all of the relevant factors and make any certain predictions, then do we throw our hands in the air and give up? We could, but often that is not a good option.

So, what then are we to do? Should we accept that it is difficult to predict the outcome with any certainty? Or should we try to estimate the relative frequency of certain outcomes?

Can we do this, can we estimate the relative frequency of certain outcomes? And how is this useful?

Here we want to estimate how often certain outcomes will happen relative to others. If the process occurs this many times, how often do we estimate we would get this result or this other result?

In other words: we want some way to determine how likely something is to happen. Is it very unlikely or quite likely to happen? Should we expect it to happen often or not very often? This is what probabilities will help us estimate.

This is all probability is, an estimate of the likelihood of a given event to occur. That is, how often a given even is expected to occur.

This helps us estimate whether we should expect something to happen in a given instance or whether we should expect it not to. As well as to estimate how often a given event might happen.

Since we cannot keep track of all the factors that determine the outcome of certain phenomena, probabilities help us deal with uncertainties. We might not be able to account for everything and predict the outcome with much certainty, but we can estimate what the results might be.

This can help us determine which results to expect and which not to expect and how often those results might happen.

This the proper role of probability in the sciences: dealing with uncertainties typically caused by our inability to track complicated or unpredictable phenomenon.

Say we find it difficult to predict the movements of an electron. We do not understand how to predict precisely where it will be two seconds from now.

However, we do know certain things about electrons. We know enough to predict that it is likely it will be somewhere in this area here. It is not likely to be in these other areas. While we cannot be certain where it will be, we at least have some ability to predict its behaviour and we might be able to do something useful.

Or take the weather. It is very hard to be certain what the weather will do days from now. But we can understand meteorology well enough to be fairly certain that on some days it will most likely rain. Or that rain is unlikely. We might not be certain and we might be wrong, but we know enough to advise people that they should prepare for these outcomes, as they are quite likely.

probability

The weather is a very complicated thing. We need a lot of very, very complicated math to predict probabilities here. And we know how often it can be wrong ….

In all these cases we are dealing with uncertainty and allowing ourselves to have some understanding of what to expect when faced with uncertainty. We might not be certain, but we know enough to say something about what is happening.

Now let us get into something more controversial.

In quantum mechanics, there is the concept of probability. But it is not treated as a method of dealing with uncertainty and making predictions about possible outcomes. There it is treated as … something else.

In quantum mechanics, the behaviour of particles is said to exist in some kind of indefinite limbo state until observed. Particles are neither here nor there but in a superposition of positions. They do not have any definite momentum and so forth. Such properties take singular, definite values only when they are observed.

One might expect that they take some definite value according to some causal mechanism. However, that is not the case, not according to quantum mechanics. Once a particle is observed, it is said that the “probability waveform” of the particle collapses and then each property takes on a definite value.

In other words, particles do not have any definite nature. They are treated as things with no definite nature. As something not fully real.