Probability theory is a branch of mathematics that studies random events, their properties and operations. In this article we will look at its definition, fundamentals and applications.
What is Probability Theory?
Probability theory uses random variables and probability distributions to mathematically evaluate uncertain situations. The concept of probability is used to assign a numerical description to the likelihood of an event occurring. Probability can be defined as the number of favourable outcomes divided by the total number of possible outcomes of an event.
Definition of probability theory
Probability theory is the area of mathematics and statistics that deals with the determination of probabilities associated with random events. There are two main approaches to the study of probability theory: theoretical and experimental. Theoretical probability is defined on the basis of logical reasoning without experimentation. In contrast, experimental probability is determined on the basis of historical data by conducting repeated experiments.
An example of probability theory
Suppose we need to determine the probability of rolling a dice to get the number 4. The number of good outcomes is 1. The possible outcomes of the dice are {1, 2, 3, 4, 5, 6}. It follows that there are 6 outcomes in total. Thus, the probability of 4 at rolling the dice, using probability theory, can be calculated as 1 / 6 ≈ 0.167.
Basics of probability theory
We can understand this area of mathematics by using a few basic terms directly related to probability theory:
- Random experiment
A random experiment in probability theory is a test that is repeated several times to produce a well-defined set of possible outcomes. Flipping a coin is an example of a random experiment.
- Sampling space
The sample space can be defined as the set of all possible outcomes resulting from a random experiment. For example, the sampling space of a symmetrical coin toss (fair coin), the sides of which are heads and tails.
- The event
Probability theory defines an event as the set of outcomes of an experiment that forms a subset of the sample space.
Examples of events are:
1. Independent – those that are unaffected by other events are independent.
2. Dependent – those influenced by other events.
3. Mutually exclusive – those events that cannot occur at the same time.
4. Equally likely – two or more events that have the same chance of occurring.
5. Exhaustive – are events that are equal to the sample space of the experiment.
- Random variable
In probability theory, a random variable can be defined as a quantity that takes on a value in all possible outcomes of an experiment.
There are two types of random variables:
1. A discrete random variable – takes on exact values such as 0, 1, 2…. It is described by a cumulative distribution function and a probability mass function.
2. Continuous random variable – a variable that can take an infinite number of values. A cumulative distribution function and probability density function are used to characterise this variable.
- Probability
We can define probability as the numerical probability that an event will occur. The probability that an event will occur always lies between 0 and 1. This is because the number of desired outcomes can never exceed the total number of outcomes of an event. Theoretical probability and empirical probability are used in probability theory to measure the chance of an event occurring.
Probability formula P(A): the number of favourable outcomes for A divided by the total number of possible outcomes.
- Conditional probability
A situation in which the probability of an event occurring is to be determined, while another event has already occurred.
Denoted as P(A | B).
- Expectation
The expectation of a random variable X may be defined as the average of the results of an experiment carried out repeatedly. The expectation is denoted as E[X]. It is also known as the mean value of a random variable.
- Dispersion
The variance is a measure that shows how the distribution of a random variable varies with respect to its mean. Dispersion is defined as the root mean square deviation from the mean value of a random variable. It is referred to as Var[X].
- The probability theory distribution function
A probability distribution or cumulative distribution function is a function that models all the possible values of an experiment using a random variable. The Bernoulli distribution and the binomial distribution are examples of discrete probability distributions. For example, the normal distribution is an example of a continuous distribution.
- Mass probability function
A mass probability function is defined as the probability that a discrete random variable will be exactly equal to a particular value.
- Probability density function
A probability density function is the probability that a continuous random variable takes many possible values.
Probability theory formulas
There are many formulas in probability theory to help calculate the various probabilities associated with events.
The most important formulas are:
1. Theoretical probability: Number of favourable outcomes / Number of possible outcomes.
2. Empirical probability: Number of times an event occurs / Total number of trials.
3. Rule of addition: P(A ∪ B) = P(A) + P(B) – P(A∩B), where A and B are events.
4. Complementarity rule: P(A’) = 1 – P(A). P(A’) means the probability that,
that the event will not occur.
5. Independent events: P(A∩B) = P(A) ⋅ P(B).
S. Conditional probability: P(A | B) = P(A∩B) / P(B).
7. Bayes theorem: P(A | B) = P(B | A) â‹… P(A) / P(B)
T. Mass probability function: f(x) = P(X = x)
9. Probability density function: p(x) = p(x) = dF(x) / dx, where F(x) is the cumulative distribution function.
10. Expectation of a continuous random variable: ∫xf(x)dx, where f(x) is MFV (Mass Probability Function)
11. expectation of a discrete random variable: ∑xp(x), where p(x) is FFT (Probability Density Function)
12. Variance: Var(X) = E[X2] – (E[X])2
Applications of probability theory
Probability theory is used in many fields and helps to assess the risks that are associated with certain decisions. Some of the areas where probability theory is applied are:
- In the financial industry, probability theory is used to create mathematical models of the stock market to predict future trends. This helps investors invest in the least risky assets that give the best returns.
- In the consumer industry, probability theory is used to reduce the probability of failure in product development.
- Casinos use probability theory to develop gambling games to maximise their profits.
Practical assignments
Example 1: When two dice are rolled, what is the probability that
that the sum of the dice is 8?
Solution: When two dice are rolled, there are 36 possible outcomes. There are 5 favourable outcomes to get the sum equal to 8.
[(2, 6), (6, 2), (3, 5), (5, 3), (4, 4)]
Using the formulas of probability theory,
Probability = Number of favourable outcomes / total number of possible outcomes = 5 / 36
Answer: The probability of getting the sum of 8 when rolling two dice is 5 / 36.
Example 2: What is the probability of drawing a queen card from the deck?
Solution: A pack of cards has 4 suits. Each suit consists of 13 cards.
Thus the total number of possible outcomes = (4) * (13) = 52.
There can be 4 queens, one of each suit. Hence the number of favourable outcomes = 4.
Card probability = 4 / 52 = 1 / 13
Answer: The probability of getting a queen from a pack of cards is 1 / 13
Example 3: Out of 10 people, 3 bought pencils, 5 bought notebooks, and 2 bought both pencils and notebooks. If a customer bought a notebook, what is the probability that he also bought a pencil?
Solution: Using the concept of conditional probability, P(A | B) = P(A∩B) / P(B).
Let A be the event when people buy pencils and B be the event when people buy notebooks.
P(A) = 3 / 10 = 0.3
P(B) = 5 / 10 = 0.5
P(A∩B) = 2 / 10 = 0.2
Let’s add these values to the formula above, P(A | B) = 0.2 / 0.5 = 0.4
Answer: The probability that the buyer bought a pencil, assuming he bought a notebook, is 0.4.
To summarise:
- Probability theory is a branch of mathematics that deals with the probabilities of random events.
- The concept of probability explains the possibility of an event occurring.
- The value of probability always lies between 0 and 1.
- In probability theory all possible outcomes of a random experiment make up the sample space.
- Probability theory uses important concepts such as random variables and cumulative distribution functions to model a random event. This also includes the definition of the various probabilities involved.
