# Two different types of probability

• disregardthat
In summary, frequentists and Bayesians have different approaches to assigning probabilities. Frequentists base probabilities on the relative frequency of outcomes in repeated experiments, while Bayesians assign probabilities to any statement based on an individual's degree of belief. Bayes' theorem is used to update one's degree of belief in a statement when new evidence is obtained. However, this does not necessarily mean that all probabilities are interpreted as degrees of belief, as there is room for a pluralist approach.
disregardthat
This is the introductory definitions of two different types of probability:

Frequentists talk about probabilities only when dealing with experiments that are random and well-defined. The probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency "in the long run" of outcomes.[1]

Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved. Probability, for a Bayesian, is a way to represent an individual's degree of belief in a statement, given the evidence.

I am only familiar with the first concept, probability based on the empirical evidence (for example a statiscial survey may help approximate the probability of a person holding a certain opinion), or determined with knowledge of all factors which can affect the outcome (for example a thought experiment by flipping a perfect coin, e.g. the unit disc thrown on a plane).

Bayesian probability states that there is a certain probability to every statement. I find this hard to justify, given that many statements is either true or false. For example, how can one determine the probability that there is 4 apples in a given basket? The statement becomes: "A: The basket b contains 4 apples". Do bayesian probability state that this true or non-true statement has a probability value attached to it, so that $$0 < P(A) < 1$$ based on a degree of belief? $$P(A)$$ should in my opinion be equal to either 1 or 0, determined by the truth of the statement. The probability that a basket chosen arbitrarily contains 4 apples is another thing. The statement becomes: "B: An arbitrarily chosen basket contains 4 apples". This, however does have a probability value $$0 \leq P(B) \leq 1$$ attached to it. I can agree that bayesian probability makes sense accompanied with extensive statistical data, but to point to a certain case which is either true or false - isn't it formally wrong to assign a probability value to it?

The above may be hair-splitting (in a practical sense), but my problem comes when bayesian probability is expanded to statements regarding the truth of undeterminable events, such as the existence of ghosts, angles, etc. When one places a probability value to a statement that is by definition undeterminable, how can this be justified by the degree of one's belief?

Last edited:

My understanding was that Bayesian is useful for looking at probablilty the other way around.
Classical stats says if you know there are 3 green apples in a basket and 1 red apple - what's the probability of pulling out a red one.
Whereas Bayesian is "if I pull out a red apple, what is the most likely proportion of red apples in the basket"
That's why it's useful in science, given this partial data - what's the likelyhood that my model is correct.

Last edited:

Yes, I agree with you on that. Statistical models is very useful in physics and mathematics. But formally, one should keep the two apart. Frequentism and Bayesian probability is in my opinion too different to be mixed with each other. I am speaking in definitions here, and not how they apply to theories and practical problems.

However, my main point was that I cannot understand how one can justify to make a claim about something that is by definition undeterminable based on a degree of belief, however that may be defined.

According to Bayesians, Bayes' theorem can be used to calculate how a rational agent should update her degree of belief in some proposition A, given some new evidence B.

It says:

P(A|B) = P(B|A)*P(A)/P(B)

The probabilities here are usually interpreted as subjective "degrees of belief".

In plain English, this means that your degree of belief in A after obtaining evidence B depends on your degree of belief that B would obtain given A, your prior degree of belief in A, and your prior degree of belief in B.

In even plainer English: evidence that would be expected given the truth of your theory, but would be surprising otherwise, is great confirmation of your theory when it turns up.

Of course the theorem itself is an analytic truth. The novelty is in applying it as an account of how we rationally update our degrees of belief. I think it's a very good account indeed.

It would be wrong to suppose that someone who likes Bayesian confirmation theory also interprets all probabilities as degrees of belief. Maybe some people do, but I think there is plenty of room for a pluralist approach. If I say "It's probably about 11pm" I'm talking about a degree of belief. If I say "This nucleus has probability 0.1 of decaying in the next minute" I may well be talking about some kind of objective physical propensity.

To answer your original question, Bayes' theorem can't tell you what degree of belief to assign to any claim at face value. It can only tell you how to update your belief when evidence comes along.

Last edited:

## 1. What is the difference between theoretical and experimental probability?

Theoretical probability is the likelihood of an event occurring based on mathematical principles and assumptions. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Experimental probability, on the other hand, is based on actual data from observations or experiments. It is calculated by dividing the number of times an event occurs by the total number of trials.

## 2. How are the two types of probability used in real life?

Theoretical probability is often used in games of chance, such as rolling dice or flipping coins, to predict the likelihood of a certain outcome. It is also used in statistics and research to make predictions and draw conclusions. Experimental probability is used in scientific experiments to test hypotheses and in business to make decisions based on collected data.

## 3. Can theoretical and experimental probability be the same?

Yes, in some cases, theoretical and experimental probability can be the same. This occurs when the actual outcomes closely match the predicted outcomes based on mathematical principles. However, in most cases, there will be some slight differences between the two types of probability due to the randomness and variability of real-life situations.

## 4. How can the two types of probability be influenced?

Theoretical probability is influenced by the assumptions and principles used to calculate it. It can also be influenced by external factors, such as changes in the sample space or the occurrence of dependent events. Experimental probability can be influenced by the design of the experiment, the size of the sample, and any extraneous variables that can affect the outcome.

## 5. Which type of probability is more accurate?

Neither theoretical nor experimental probability is inherently more accurate than the other. The accuracy of each type depends on the quality and reliability of the data used and the assumptions made. In general, theoretical probability is more precise and predictable, while experimental probability is more reflective of real-life situations.

• Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
• Calculus and Beyond Homework Help
Replies
31
Views
3K
• Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
• General Math
Replies
2
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
45
Views
3K
• Quantum Interpretations and Foundations
Replies
14
Views
2K
• Quantum Interpretations and Foundations
Replies
37
Views
2K
• Calculus and Beyond Homework Help
Replies
6
Views
1K
• Cosmology
Replies
96
Views
9K