A philosophical musing concerning probability

Factual Questions
1

Hi, first post here, have been lurking and learning for ages, what a wealth of knowledge on this site, I am in total awe …

Anyway, I have a question which has bugged me for many years, and it is concerned with the whole concept of probabilty . I figure the expert mathematicians and philosophers here (into neither of which category I fit) may be able to assist.

Now, my problem concerns when people talk about the “probability of an event happening”.

On the face of it, this would seem a harmless enough mode of expression, but when you actually start to analyse it, it seems to me that there really isn’t such a thing as the “probability of an event happening” what you have is the “probability of you being right if you forecast a certain outcome of the event” and that the probability is dependent on the amount of information which you possess about that event.

So, for example, tossing a coin is an event about which you cannot have any information (unless you know that the coin is biased) but that isn’t to say that the information isn’t there, it just means we don’t have access to it. It is at least conceivable that given enough sophistication in measurement of all variables, it would be possible to forecast the eventual outcome of any indivudual coin toss with a significantly higher success rate than 50%. Not that anybody is ever going to do this, but theoretically at least, it is conceivable.

Going up a step, consider a roulette wheel… Now, imagine a situation where a skilled croupier is able to throw the ball into one physical half of the wheel with greater than 50% success rate. When he chooses to do this, the probability of the ball landing on one of the numbers in his chosen half of the wheel becomes greater than 50% but only he is aware of this because he has more information. So, his probability is different from the punters’ probability.

Finally, consider horse racing. In a 6 horse race, for the sake of argument, your chance of selecting the eventual winner at random, knowing nothing about the form of the runners, is 5/1. If, however, you are a devoted form student, you may well be able to nominate the winner of such a race with a far greater accuracy so that **your **probability for the various runners could vary from say, 1/1 out to, say, 66/1. If you are the trainer of one of the runners, then your probability could be a whole lot different again.

Hence my contention that probability is not a property of the event itself, it is a function of the amount of information which you have about the event. So am I right, or a meringue ?

2

What you’re describing is subjective probability, and there are many situations where that’s the most reasonable interpretation of probabilities. Google that phrase for much more reading.

3

Absolutely (especially in terms of horse racing ) but my contention is that ***all ***probabilities are subjective probabilities, they only become considered as ***objective ***probabilities simply because it is impossible to actually access sufficient information to make them otherwise, although the information is there if we had the tools to measure it.

I believe that Laplace took a similar approach to the question …

4

You are correct: probability is not a property of a single event or situation. Probability can only have meaning in the context of an “ensemble”, a repetition of events or copies of a situation. The probability of an event having a particular outcome is just the fraction of events in the ensemble that have that outcome.

Of course most of the time we can’t or don’t actually want to do the repetitions. We instead imagine the repetitions in a thought experiment, and the probability very much depends on how we “design” the thought experiment. In particular, it depends on how much ignorance we build into the experiment.

For instance: flipping a coin. It’s easy to imagine the thought experiment of flipping the coin 100 times and counting the heads. The “ignorance” we design in is that we will not specify the force, angle, height, etc of the flips and we are “ignorant” of any bias in the coin. Then we can deduce a probability of 50% heads. However, we could also imagine flipping the coin with a highly-repeatable machine from a known height in a stable environment, we might know enough to say that the coin will land heads 90% of the time.**

For a race between six horses we could, of course, run the race 100 times (or imagine doing so). But what ignorance are you building in? If you imagine you know nothing about any of the horses beforehand or of the conditions of the track, then your experiment is to run 100 random sets of horses against each other on 100 randomly selected horse tracks. In that case we can pretty we well guess that horse #6 will win one sixth of the time. However if we build in the knowledge that horse #6 is lame or that lane 6 contains a pit trap, then we can deduce a probability of less than one-sixth.

What is the probability, say, of the existence of Earth? If you asked me that, you would need to tell me ensemble: What is it that is it that could be repeated that may or may not lead to the existence of Earth? Without specifying that, the question is meaningless.

When probability questions spark long, rambling debates, it’s a sure sign that the ensemble is not sufficiently specified and thus ambiguous. Different readers imagine different “experiments” and thus deduce different probabilities. This is exactly what happens in the disputes over the Monty Hall problem and the 2nd-child problems.

5

OP is correct and makes a point which is often overlooked. In many common situations, probability estimates describe one’s own ignorance.

For example, Alice is playing craps, makes her come-out throw but because of obstacles cannot see the dice she’s just thrown. Bob can see one die – it’s a Five. Carol can see the other die – an Ace. Doug sees both dice.

What is the probability Alice’s Come bet will win?

Alice has the probability at 49.293% – the same as it was before she cast the dice; her uncertainty hasn’t changed.

Bob figures the probability at 60.71% and would be happy to place a side bet based on that. Carol figures 36.46%. Doug figures 45.45%.

All four different probability estimates are correct – they reflect different people’s ignorance.

Even probabilities of future events may be affected in this way. In fact, this simple example shows that, as this dice cast does not conclude the Come bet. (Someone will be along to say that quantum physics probabilities are an exception but (a) these are irrelevant in most everyday cases and (b) such “probabilities” are spooky and perhaps not fully understood anyway.)

6

I run into this all the time in biology. For instance, if we know that two people heterozygous for a disease allele have a child, we can say that the odds of the child being heterozygous are 1 in 2. However, if we then add the information that the child doesn’t have the recessive disease, then the odds change to 2 in 3. A lot of students have a hard time wrapping their heads around the fact that probability changes based on how much information we have.

7

Both septimus and Smeghead are describing conditional probabilities, which are consistent with both subjective and frequentist notions of probability.

8

Sounds like OP is getting close to describing the sorts of thing Bayesian decision theory would deal with.

9

Sure you are right.
From a purely mathematical point of view, probability is just a mathematical function, so it is an abstract concept which doesn’t relate to the real world.
But in a real world application, yes, probability is generally used to express the amount of information that you have about an event.

Having done Bayesian analyses for a few years, I can confirm that this is exactly the type of point of view that Bayesian people would adopt.