Theoretical probability is a probability in which the probability of an event is calculated based on anticipated outcomes and not on actual experiments. In statistics and mathematics, probability is the likelihood of an event taking place. There are various kinds of probability, such as experimental, theoretical, axiomatic, and subjective. In this article, we will discuss theoretical probability by understanding its definition, formula, and real-life examples. Whether you are getting ready for U.S. competitive exams such as the SAT, ACT, GRE, or GMAT, it is crucial to master probability concepts to enhance your problem-solving skills and overall performance.

Theoretical Probability

The likelihood of a particular event is what we learn in probability. Moving forward to what is a theoretical type of probability in mathematics.

Theoretical probability definition: In a certain type of experiment where we are already having a predefined set of assumptions, the repetition can be avoided in such cases. The reason being those assumptions help estimate the exact or the theoretical probability of the given event.

Example: In tossing a coin both head and tail can be said to have a predefined probability of 1/2 or 0.5. The reason being that is there are only two outcomes that can happen.

In the same manner when you roll a fair dice the possible outcomes are 1,2,3,4,5, and 6.

Formula of Theoretical Probability

Theoretical probability as the name suggests deals with the theory behind probability. Such a probability can be estimated either by applying logical reasoning or by using the formula. In the previous header, we read the definition moving forward to the theoretical probability formula.

Theoretical Probability Formula = Number of favourable results / Total number of likely outcomes.

Theoretical Probability Distribution

The purpose of theoretical probability distributions is to provide mathematical models for describing the likelihood of different outcomes in random experiments. They are used in probability theory and statistics to analyse uncertain events. There are two main types of theoretical probability distributions.

Discrete probability distribution

Discrete probability distribution is used when the possible outcomes are countable and distinct. It includes distributions like the binomial distribution, which models the number of successes in a fixed number of independent trials, and the Poisson distribution, which describes the probability of a given number of events occurring in a fixed interval of time or space.

In mathematical notation, the discrete probability distribution is typically represented using a probability mass function (PMF). For example, if X represents the outcome of rolling a fair six-sided die, the PMF could be expressed as \(P(X = x) = \frac{1}{6}\) for \(x = 1, 2, 3, 4, 5, \text{ or } 6\).

Continuous probability distribution

Continuous probability distribution is used when the possible outcomes form an interval. It is often applied to measurements involving time or distance. The normal distribution is a well-known example of a continuous probability distribution. It is characterized by its symmetric shape and is commonly used to model naturally occurring phenomena, such as the heights or weights of individuals in a population.

The continuous probability distribution is usually described by a probability density function (PDF). The PDF of the normal distribution, denoted by \(f(x)\), specifies the relative likelihood of different outcomes. Its formula is:

\(f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\)

Here, \(\mu\) represents the mean or average value of the distribution, and \(\sigma\) represents the standard deviation, which measures the spread of the data.

Understanding and using theoretical probability distributions are valuable in predicting, inferring, and solving real-world problems in various fields, including finance, engineering, and social sciences.

Difference between Theoretical and Experimental Probability

Theoretical Probability Solved Examples 

The theoretical distribution of probability deals with the theoretical assumption to find the occurrence of an event. This states such an event does not require experimental data. On the other hand, going as per the formula it is defined as the ratio of the number of favourable results to the number of possible results. Let us step towards some solved examples related to the topic.

Solved Example 1: A fair coin is tossed once. What is the probability of head/tail?

Solution: According to the theory, in tossing a coin, it is pre-stated that there will be two outcomes(namely head and tails). Therefore the number of possible outcomes is 2.

Due to this:

P(Head) = 1 / 2 = 0.5

P(Tail) = 1 / 2 = 0.5

Solved Example 2: Obtain the probability of getting 5 once rolled on a fair die.

Solution: The possible outcomes of rolling a die=6 i.e. 1, 2, 3, 4, 5 and 6 as per the knowledge of the circumstances.

As per the given question, the probability of getting 5 on the face of dice = 1

Hence, P(5) = 1 / 6

Solved Example 3: If in a box there are 4 blue and 8 green marbles. Determine the probability of choosing a green marble.

Solution: Using the theoretical probability formula.

Theoretical Probability = No. of favourable results / No. of probable results.

No. of favourable results = 8

Number of probable results = 4 + 8 = 12

P(green marble) = 8 / 12

We hope this piece of writing has been useful in solidifying your knowledge of theoretical probability and in making you exam-ready. Stay tuned for more tips and information on key mathematics concepts that are applicable for U.S. competitive exams like the SAT, ACT, GRE, and GMAT. Don't forget to check out practice tests and resources to test your knowledge and improve your confidence for these tests.