Introduction

Probability, simply put, is the chance whether a specific event will occur or not. Based on the number of possible outcomes, the probability of a specific event predicts how favorable that event can be or what is the chance of having that event. Many events are impossible to predict with 100% accuracy, but the likelihood of their taking place may be predicted. All probabilities are expressed as a number between zero and one.

A probability of zero indicates that the event in question is impossible. If an event has a probability of one, it is sure to happen. If an event has a probability between 0 and 1, it indicates how likely it will occur. 

The closer the probability is to zero, the lower will be its chance of happening. The closer it is to one, the more will be its chance of happening. The sum of all probabilities for a given event always equals one.

Theoretical Probability 

Theoretical probability, simply put, theorises probability. In other words, it is a formal representation of its concepts. It includes terms that may be studied independently of their meaning. The laws of mathematics and logic are used to handle these formal concepts, and any outcomes are interpreted into the problem area.

The Kolmogorov formulation and the Cox formulation are two examples of successful attempts to formalise probability. Sets are interpreted as events in Kolmogorov’s formulation (see also probability space), and probability is measured on a class of sets. 

Probability is treated as a primitive (i.e., it is not further examined) in Cox’s theorem, and the focus is on creating a consistent assignment of probability values to propositions. Except for technical differences, the laws of probability are the same in both theories.

Other methods for quantifying uncertainty exist, such as the Dempster–Shafer theory or possibility theory, but they are fundamentally different and incompatible with commonly accepted probability principles.

Probability

In a given experiment, probability measures the possibility that a specific event will occur. Probability can be interpreted as a number between 0 and 1, with 1 indicating certainty and 0 indicating impossibility. The greater the likelihood of an occurrence, the more likely it will occur.

Conditional Probability 

The possibility of an event or outcome occurring based on a preceding event or outcome is known as conditional probability. The updated probability of the subsequent or conditional event is multiplied by the probability of the preceding or conditional event to get conditional probability.

Defining Conditional Probability Formula

The conditional probability formula is defined by the multiplication of the preceding event’s probability by the conditional or succeeding event’s probability. Basically, conditional probability focuses on the occurrence of a single event’s probability depending on the preceding event’s probability. 

Independence 

Two events are said to be independent of each other if the probability that one event occurs in no way affects the probability of the other event occurring. 

Conditional independence 

In conditional independence probability theory, it talks about the situations where an observation is completely redundant or unrelated during the evaluation of a hypothesis’ certainty.

Example: 

Let us consider ‘a’ as the given hypothesis where ‘b’ and ‘c’ are observations. In this scenario conditional independence will be represented as: P(a|b,c) = P(a|c). Here, P(a|b,c) defines the probability of ‘a’ given both ‘b’ and ‘c.’ As the probability of a given ‘c’ has the similar probability as ‘a’ given both ‘b’ and ‘c.’ The equality obtained here shows that ‘b’ has no role in determining the certainty of ‘a.’ In this scenario ‘a’ and ‘b’ will be conditionally independent of given ‘c.’ It will be demonstrated as: (a || b|c). 

Conditional independence depends on the nature of the third event. If you roll two dice, one may assume that the two dice behave independently of each other. Looking at the results of one die will not tell you about the result of the second die. (That is, the two dice are independent.)

Distribution of probabilities

It is a mathematical function that translates the probabilities of all conceivable outcomes of a random experiment. Whether the random variable X is discrete or continuous is determined by the random variable X.

Example

Let us consider the tossing of a coin. Here, we take x as a random variable that represents the heads’ count that is obtained. So, here we will find the distribution of probability for x. 

Considering x’s value as 0, 1, and 2. We can take possibilities such as: 

• No number of head counts
• One tail and one head
• Both coins have the heads

Here, we can describe distribution of probability as following: 

P(X=0) = P(Tail+Tail) = ½ * ½ = ¼

P(X=1) = P(Head+Tail) or P(Tail+Head) = ½ * ½ + ½ *½ = ½

P(X=2) = P(Head+Head) = ½ * ½ = ¼

Conditional Probability Distribution 

The conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a certain value in probability theory and statistics. In some circumstances, the conditional probabilities may be written as functions containing the undetermined value x of X a parameter. A conditional probability table is often used to illustrate the conditional probability when both X and Y are categorical variables. The conditional distribution of a random variable differs from its marginal distribution, which is its distribution without regard for the value of the other variable.

The conditional density function is the probability density function of the conditional distribution of Y given X if it is a continuous distribution. The conditional mean and conditional variance are two terms used to describe the attributes of a conditional distribution, such as the moments.

• Formula

For all x, use the formula p(x) = P(X=x). The discrete random variable X is known as the probability mass function.

Examples

Conditional probability: p(A|B) is the chance of event A happening if event B happens. What is the likelihood that you drew a red card (p(four|red))=2/26=1/13 if you drew a red card? So there are two fours out of the 26 red cards (given a red card), so 2/26=1/13.

Conclusion 

People employ probability and odds in various scenarios, such as deciding how to dress for the weather, whether or not to buy a stock, and how much to risk when gambling. Probability assists people in determining which options are safe and which are risky. Of course, having a thorough understanding of probability makes this process much easier. We can learn about the likelihood of future events and plan accordingly by knowing about probability.

Probability assists people in determining which options are safe and which are risky. Of course, having a thorough understanding of probability makes this process much easier. We can learn about the likelihood of future events and plan accordingly by knowing about probability.