# How does one compare the probability of an outcome vs. an event?

Suppose Adam guesses a number between 1 and 10 from a random number generator. Suppose Bethany guesses a number between 1 and 100 from a random generator.

The probability of Adam guessing the correct outcome is 1/10, whereas it’s 1/100 for Bethany.

Something about this seems to create the implication that it’s “harder” for nature to create the first event instead of the second. But is this misconstruing probabilities?

What is the probability of the event of Adam guessing the correct outcome occurring vs. the event of Bethany guessing the correct outcome? It is not as if there were 10 events with equal likelihood in a hat where one of them was Adam picking the correct number that nature decided to choose from. How does one then compare probabilities of events?

If the number generator is truly random, the probability of it choosing a particular number for Adam is 1/10 and for Bethany 1/100, but this is different than stating:

"The probability of Adam guessing the correct outcome is 1/10, whereas it’s 1/100 for Bethany".

Why? Because there might be any number of factors which make it more likely that Adam and Bethany will choose a particular number over others. For example, perhaps Adam heard a particular number repeated on the radio earlier that day, or perhaps Bethany's favourite number is 27. This in turn impacts the odds that they will choose other numbers, although we are often unaware of such biases when arriving at probabilities.

It is important though to realise that we use probabilities most often to make predictions about the future, and that we shouldn't confuse the conclusions we draw about those probabilities with the notion that these probabilities actually exist.

When an event happens, we have no way of knowing for certain that it could ever have happened otherwise. All we have to go on is a sample size of 1. You might postulate the odds of a 47 year-old, chain-smoking, alcoholic habitual drunk driver dying on any given day to be about 1/18,647,236, but if that person actually dies, there's nothing we know about probabilities that suggests they could have died at any other time. (Some might argue the unpredictability of quantum processes might be an exception to this idea. I don't know enough about quantum mechanics to make a confident statement either way. Someone else might clarify).

As for:

A number being chosen at random (if a random number machine is truly possible), upon being 'asked' to do so is inevitable (barring malfunction/interference). The 1/100 generator may have to do more 'work' at some level in order to select a number (if, for example, it works via a digital 'shuffling' of the numbers prior to selection), but I don't think this is what you're referring to. The 'event' would still, as far as we can determine, have been inevitable; perhaps a part of a causal chain of prior events. From this perspective, comparing the probabilities of events that have occurred might be deemed either easy (if you assert that all events have a probability of 1), or impossible (if we admit we can't know if randomness truly played a part).

• I also wonder the same as to whether something could have happened otherwise. Quantum mechanics does point to indeterminism being possibly true. But indeterminism does not disprove inevitability. The only way to definitively disprove inevitability would be to rewind time and see what happens which is impossible Aug 16 at 16:04
• Agreed (albeit with no physics background). Aug 16 at 16:18
• What makes you suspect that a random number machine might not be possible? In reality we have all kinds of random number machines, we have had dice for thousands of years. The possible biases in Adam's and Bethany's behaviour make no difference, if they are trying to guess a random result. We are still looking at the probability by which the random machine will match the non-random guess. Aug 17 at 11:11
• @PerttiRuismäki. I take your point about dice, although there's also a strong case to make that the way they land are determined entirely by the conditions of the throw/environment. In relation to RNMs, I just wasn't confident that the algorithms they use are totally random, ie. that a pattern would theoretically be at some scale possible to detect. I fully acknowledge this might be wrong though. Aug 17 at 11:21
• @Futilitarian But the conditions of the throw are the random part of the equation. You cannot know how to throw the die to get a specific result and even if you knew exactly you could not perform the throw with sufficient accuracy. True random number generators don't use an algorithm, they have to extract random values from a naturally stochastic process (=noise). Random is the opposite of deliberate. Adam & Bethany's guesses are not random, they are deliberately chosen. Aug 17 at 13:08

There is arguably no such thing as randomness in the macroscopic world; it has not yet been completely ruled out that the quantum world is non-random (Bell's theorem doesn't cover all possibilities). So what is "randomness"?

Randomness, or more formally probability and statistics, is a means of constructing a phenomenological model of reality, i.e. a model that can be used to predict what happens, but without necessarily explaining how or why it happens. A model of reality and reality are not the same thing, so

Something about this seems to create the implication that it’s “harder” for nature to create the first event instead of the second. But is this misconstruing probabilities?

is indeed misconstruing probability as it is conflating a model with reality (nature). When you say:

"Suppose Adam guesses a number between 1 and 10 from a random number generator. ... The probability of Adam guessing the correct outcome is 1/10, ...

That is not a direct statement about reality, instead you have constructed a simple statistical model of Adam's behaviour, that assumes that Adam's guesses will be an equiprobable draw from the numbers from 1 to 10. However, as @futilitarian correctly points out, Adam being a human being is very unlikely to make equiprobable guesses. Apparently, most people are likely to choose 7 more than any other number. So the model is likely to be "wrong" in the sense that it doesn't match reality, but that is fine because all models are necessarily "wrong" in that sense:

All models are wrong, some are useful.” [G. E. P. Box]

Models are necessarily simplifications of reality. If we could fully understand reality in complete detail, we would have no need of a model. Models are always simplifications of reality, they ignore factors. Statistics and probability allow us to make models of regularities in the behaviour of a complex system (such as coin flips, guessing numbers, which horse will win a race) by ignoring initial conditions (in chaotic systems) or ignoring all of the variables and summarising them (e.g. statistical mechanics). Whether a model is useful depends on whether it provides a good enough approximation to reality for the purpose at hand. Physical models (e.g. predicting planetary orbits) are no different in that respect - the model is "wrong",,= it is not reality and may or may not provide a useful approximation to reality.

In this case all we can say is that according to the model, the first event is more likely than the second. Any conclusion you want to draw about reality "nature" requires you to demonstrate that the model is "useful" and is contingent on that assumption.

"Something about this seems to create the implication that it’s “harder” for nature to create the first event instead of the second. But is this misconstruing probabilities?"

I can't see anything that implies that. If "nature" is well described by the model, then it is easier to create the first event as it is ten times more likely.

What is the probability of the event of Adam guessing the correct outcome occurring vs. the event of Bethany guessing the correct outcome?

Unfortunately you are obfuscating your intended meaning by redefining existing terms used in probability to mean something else. An "outcome" is any specific way in which a trial can work out, so if you roll a six sided die, then the outcomes are 1, 2, 3, 4, 5 and 6. In probability an "event" is any specific set of outcomes, for instance the event "roll and even number" is the set of outcomes 2, 4 and 6.

"It is not as if there were 10 events with equal likelihood in a hat where one of them was Adam picking the correct number that nature decided to choose from"

That is because it is an assumption of the model, not of reality. The model is based on the assumption that this is the probability of the event if it was an independent and identically distributed sample from a population of such events. Frequentists statistics often do this by having a notional/fictitious population of die rolls/coin flips that the observations come from. This is because they fundamentally cannot assign a probability to a specific event (as the event doesn't have a long run frequency, it only happened once and turned out the way it did). If you want to assign a probability to a specific event then it has to be a Bayesian probability, which can represent a belief, state of knowledge, plausibility depending on which school of Bayesianism is followed.

How does one then compare probabilities of events?

You have two choices: (i) Be a Bayesian and perform the comparison directly, or (ii) be a frequentist and substitute a comparison of (possibly fictitious) populations from which the event can be regarded as a sample. Both frameworks are useful, but you do need to understand them (the frequenist frame work has a lot of subtleties that are rarely communicated to students) and be able to decide which framework is most appropriate for the question you want answered.