probability is defined as the expected value of a random variable. In other words, it is the frequency with which a certain property is obtained by chance or by the assumption of a relationship between probability and values. If you take a baseball and give it a normal swing, it will definitely go over a given distance in the given direction over a certain number of times. probability on the other hand is the expected value that results from the given probability distribution. probability compares the value of a normal swing with the value of other normal swings based on their distributions. So basically it is about comparing how likely a given probability distribution is to generate a value.
Probailty is one of the most important concepts in statistics but is often underrated because its significance is usually overlooked by researchers and their measurement of probability is often arbitrary and / or very uncertain. The main problem with probaility is that its definition relies on a model of the universe and on the assumption that there are random variables which can be transformed into other random variables using an observer who can determine their values. These variables can then be analyzed to produce a measure of the probability that the event occurs. This process is called “The Problem of the Event Space.”
The Problem of the Event Space states that since probabilities can be transformed, an observer cannot tell with any great degree of accuracy what the actual probability distribution for any particular event is. The trouble is that the actual distribution may have very subtle properties and it is often very hard to analyze these properties. To illustrate, suppose that we want to estimate the distribution of the number of white adults in a sample of a hundred kids who were selected at random. How can we know what the distribution will look like? We can’t just ask a bunch of random people what their probabilities are, because even if they tell you that their probability of being white is 10.5% and you find that they are all white adults, you still don’t know what the range of probabilities that you are looking at is since there isn’t anyone left to calculate it for you!
A probabilistic statistical distribution can solve this problem. It is a mathematical model that evaluates the expected value of a random variable in terms of its mean and standard deviation. The mean and standard deviation are the mathematical equations that give you the expected number of successes or failures from a uniform distribution. They are easy to understand and use in the probabilistic world. probability arises as a function of the probability density of the event. This is called “the density”.
The density of probaility is actually a measure of the “liars” in the population – those that are outside the range of “the truth” in terms of their actual probability. In the probabilistic world, the distribution of random variables follows a normal distribution due to the uniformity of the distribution. It can be visualized as the curve on a graph. If you draw a line from the point where the curve begins to form and end to where it curves out, this is the value of the random variable that you will expect to see.
In addition to the normal (Gaussian) distribution of probabilities, there exists a normal curve which can be called the log-normal probability curve. This is where the range of probability of the event space is not continuous but instead breaks down into smaller probability spaces. For instance, if the probability range is from one to two percent, then the interval between each event occurs every five percentage points. You can observe that this breaks down further into smaller intervals within the range of one to five percent. This is the log-normal probability space.
The likelihood of each event occurring lies in the joint probability distribution. This distribution follows a normal curve that is continuous over the range of the probability space. Its shape is such that it can be drawn on a disc and plotted on a graph. You have just visualized the probabilistic distribution of the random variables. It is very important to understand this concept if you are to understand the statistical distribution of the outcomes.
You may find many references to the log-normal probability or the log-normal distribution in textbooks, articles, and so on. However, learning more about the probabilities and their relation to probability spaces is more fruitful than merely reading about them. The best way to gain more knowledge about the subject is by taking the necessary computational mathematics classes or obtaining some computer programming knowledge. Alternatively, you could do a course in probability theory and study the foundations of statistics and probability courses. There are many books that explain the various concepts and give examples of real world applications relevant to decision science and management.