What is Theoretical Probability?
Theoretical probability is the probability of an event predicted through logical reasoning, a clear understanding of the situation, and a probability model—not by running an experiment. It is based on a known model and set of assumptions that define the likelihood of different outcomes. Instead of observing what happens, you use what you know about the process.
Of course, the accuracy depends on the model’s validity. If you use the wrong model, you’ll get inaccurate results!
This type of probability answers the question: “What should happen in theory according to a probability model?”
How to Calculate Theoretical Probability
All theoretical probabilities rely on a model that defines how likely each outcome is, ranging from simple to complex. These models can come from logic, physical reasoning, or mathematical structure, but they do not rely on observed data.
The most basic model assumes that all outcomes are equally likely. In these cases, theoretical probability is calculated using the classic formula:
Theoretical Probability = (Number of favorable outcomes) ÷ (Total number of possible outcomes)
This applies to familiar situations like flipping a fair coin, rolling a die, or drawing a card from a well-shuffled deck.
More advanced models allow for unequally likely outcomes. For example, a biased coin or non-uniform spinner can have defined, unequal probabilities for each outcome. As long as those probabilities are specified by the model, analysts can use them to calculate theoretical probabilities.
Theoretical probability also includes values derived from probability distributions, such as the normal, binomial, and Poisson distributions. These distributions describe how outcomes are expected to behave under certain mathematical assumptions and allow analysts to calculate probabilities for specific ranges or values.
At the most complex level, logistic regression or other predictive models produce theoretical probabilities for outcomes based on a combination of predictor variables. Even though these models are estimated using data, the probabilities they output are theoretical because they reflect the model’s theoretical expectations.
If no suitable model exists or if the process is too complex or poorly understood, experimental probability might be the better alternative by relying on observed outcomes instead.
Examples
Consider a standard deck of cards that contains 52 cards. To find the probability of randomly drawing a heart, there are 13 favorable outcomes (hearts) and 52 total outcomes (cards). The theoretical probability is:
13 ÷ 52 = 0.25, or 25%
This answer is based on logical reasoning and the assumption that the deck is complete and well-shuffled, with each card’s likelihood defined by the structure of the deck.
You can also use a probability distribution to calculate theoretical probabilities. For example, if you toss a fair coin 10 times, the number of heads you expect to see follows a binomial distribution with n = 10 and p = 0.5. The probability of getting exactly 6 heads is:
P(X = 6) = 210 ÷ 1024 = 0.205, or 20.5%
This is a theoretical probability based on the binomial model—it tells you what should happen under those defined conditions.
Another example of theoretical probability comes from logistic regression, a statistical model used to estimate the probability of a binary outcome based on one or more predictors. For instance, suppose researchers develop a logistic regression model to estimate the probability that a patient is readmitted to the hospital within 30 days. The model uses predictors such as age, length of stay, and number of prior admissions.
For a specific patient, the model outputs a predicted probability of 0.37. According to the model, there is a 37% theoretical probability that the hospital will readmit this patient. That value is not based on observed frequency but on what the model predicts under its assumptions and parameters.
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