The concept of probability holds a significant position in the realm of mathematics and extends beyond its boundaries. It plays a crucial role in decision-making, outcome prediction, and uncertainty analysis. This all-encompassing guide aims to unravel the intricacies of probability by delving into its fundamental principles and pragmatic implementations.

Probability holds significant importance within the realm of STEM education.

## 1. Introduction to Probability

A probability distribution is a table or equation that links each outcome of a statistical experiment with its probability of occurrence. It is often represented by a histogram as well. Throughout this assignment, the concept of a probability distribution can be shown with various graphs. A random variable can be classified as either discrete or continuous. If it takes on a finite number of possible values, it is referred to as a discrete random variable. A very basic example of probability is the probability of getting heads in a simple coin toss and its application in decision theory. On the other hand, a continuous random variable has an infinite number of possible values. In any case, the probability of an event is expressed by a range of values of the random variable. It is given by the sum of probabilities of all the specific events. Here is a relevant example: if one wants to know the probability of throwing darts and hitting the center of a dartboard since the chances of hitting the aim are not too bad. This scenario can be represented by a continuous random variable.

The measure of probability can be presented in either fraction, percentage, or ratio form. With the previous example of the die, the probability of getting a six could be presented as 16.67%. Probability, on occasions, appeared to be a relative frequency. The chance that a relative frequency will produce a good guide into predicting the future event has created another type of probability called frequentist probability. It states that the probability of a particular event is equal to the frequency of that event, given an infinite number of possibilities. An example can be comparing the chance of getting a head in the tossing of a bent coin and the possibility of rain on Christmas Day. We will likely keep up to date with the weather forecast for the day in question.

Probability is the chance of a particular outcome occurring. For example, if a die is thrown and one wants to know the probability of getting a six, there is only one possible outcome, so the probability is 1/6. This is an example of theoretical probability. It is called theoretical because if the experiment were to be conducted a large number of times, the resultant frequency would tend to become close to the theoretical probability. It is given by the formula: probability = (number of favorable outcomes)/(total number of possible outcomes).

Probability is an interesting branch of mathematics involving statistical rules of occurrences applied to gambles, and games, and forecasting the number of specific events. This essay discusses several different types of probability.

### 1.1. Definition of Probability

The modern computational form of probability is still in its formative stages, opening exciting avenues of applications from stochastic processes and Monte Carlo simulations to the increasingly relevant field of algorithmic inference. Yet fundamentally, probability is largely unchanged from its original inception as a useful tool in roughing out uncertainty to aid better decisions.

Given an event Ai in a sample space S, an assignment of a number pi, 0 <= pi <= 1, to the event is called a probability distribution on the sample space S. If the sample points are equally likely, the probability of an event is the ratio of the number of sample points in the event to the number of sample points in the sample space. Often the model of an equiprobable sample space is employed for ease in calculations and lack of a priori information to the contrary, though it may not be realistic about the given situation.

Probability is a measure of uncertainty of various phenomena. It is likely with the happenings of the events. The classical method is confined to situations possessing equal likelihoods of occurrences, an assumption that is not always realized in practice but is often held implicitly in the absence of a specific assigning of probabilities. The coefficient ranges from 1 (exclusive certainty that an event will occur) to 0 (exclusive certainty that it will not occur).

### 1.2. Importance of Probability in Statistics

Once data has been collected from a study, one of the basic uses of probability is to make inferences about the population being studied. That is, to use the data to draw conclusions and make decisions about the population, one must realize that the data is but a small part of the entire population, and thus there is uncertainty about whether an inference based on the data is correct. Probability is the tool that allows one to quantitatively describe the uncertainty. This is often done using an estimation procedure, where a confidence interval is constructed to contain the value of a parameter with a certain probability. Another method involves testing a specific hypothesis. Here one must realize that the data may not clearly indicate whether the hypothesis is true or false, and so a decision rule is set to either reject the hypothesis or to make no conclusion, to minimize errors in either direction. Probability is used to assess the likelihood of various outcomes of the decision rule, to ensure that it is logically consistent with the available evidence.

Alternatively, the sample size may be fixed, but the study may be designed in a manner that will improve the quality of the results. This too can involve considerations of various probabilities. For example, an epidemiological study of the effects of alcohol consumption on health may be done using telephone interviews, and a high non-response rate would decrease the accuracy of the results. The investigators might then consider the probability that a household’s participation would increase if it were telephoned again after no one answered the first time, and would base the decision about whether to make a second call on the overall cost and the perceived increase in quality.

In planning a statistical study, the probability of various potential outcomes can be used to determine what the sample size should be. This can be an effort to reduce the uncertainty of the estimates for a given margin of error.

The importance of probability in statistics is well appreciated by statisticians. There are various ways in which probability theory plays a role in statistics.

### 1.3. Basic Concepts

Random variables are used to measure the uncertain quantities of events. Consider an experiment of tossing a coin three times. The possible events are the number of heads which could range from 0 to 3 inclusive. The uncertain quantity (i.e. the number of heads) is represented by a random variable X. Random variables can be classified into two types: discrete or continuous. A discrete random variable has a set of outcomes which are separate and distinct, while a continuous random variable has an infinite set of outcomes which are not countable. In the case of the coin-tossing experiment, the number of heads is a discrete random variable.

The probability of an event is a measure of the likelihood that the event will occur. This measure is a number between 0 and 1. A probability of 0 indicates that there is no likelihood that the event will occur, and a probability of 1 indicates absolute certainty that the event will occur. An event with a probability of 0.5 is considered to be equally likely to occur or not occur. It is important to note that the possible events together must encompass the entire sample space. This means that the sum of the probabilities of all possible events is equal to 1. An event that is certain to occur has probability 1. The complement of an event A is the event that A does not occur. The probability of the complement of event A is 1 minus the probability of event A.

### 1.4. Applications of the theory

One of the simplest models of a clinical trial is the 2×2 table. Before discussing clinical trials, let’s consider the following example: in a certain disease, it is well-known that the probability of developing malignant hypertension is 0.08. It is also known that it will have no effect on patient survival rate and therefore the probability of recovering from the malignancy is 0.2. Taking these two events, define the following: A ≡ event patient receives treatment for malignancy B ≡ event patient recovery is noticed We can use a probability tree diagram to show the sums of joint probabilities of the 2 events and the total probability of event B. This result has defined the “baseline” for this particular patient’s post-career and we can do a similar tree diagram using the same initial probabilities to define the outcomes of the 2 events. Now let’s consider an RCT on malignant hypertension in the same group of patients.

## 2. Probability Formula

It follows from the rules of the addition of probabilities that if E is any event, the chances of E and of not-E together are equal to certainty; that is, P(E) + P(not-E) = 1.

Let A be any event of which we know the numerical chance, which we denote by P(A), and let B be any event: then we call the numerical chance which the relative event A ∩ B bears to the event B the chance of A on B, and we denote it by P(A ∩ B)/P(B).

We are now in a position to investigate the chance of the occurrence of A under the presumption that B is an ascertained event, i.e. we want to find P(A/B); it is therefore necessary that P(A) shall be expressible in one and only one way as a compound event using events each of which involves, and is expressible in, an event such that that P of the latter is known.

Now P(A) = P(A ∩ B) + P(A ∩ not-B), and by the supposition, this right-hand side is equivalent to P(B)P(A/B) + P(not-B)P(A/not-B), that is P(A) is equivalent to P(B)P(A/B) together with this quantity multiplied by P(not-B).

Now it is evident that the attempt to determine A by reference to B is tantamount to forming a certain event C, which is the result of drawing a ball that affects B, with the object of finding the chance of C; such a result is a change in the drawing of a ball from one urn to another, and this process is fully discussed in probabilities of error in the Theory of Logic and the article PROBABILITY.

Therefore we can say that the following method of determining P(A/B) is in all typical cases doomed to success. We form the product of both members of the equation by P(not-B) and make use of P(A ∩ B) = P(B)P(A/B); thus we have P(not-B)P(A) = P(B)P(A/B)P(not-B) + P(not-B)P(A/not-B) and hence P(A) = P(B)P(A/B) + P(not-B)P(A/not-B).