Probability of Elementary Elements

Introduction

Elementary probability gives the likelihood that an event that is defined will take place. Experts quantify it as a positive number whose existence is between 0 and 1. The implication of 0 is that the event is impossible to happen. In contrast, 1 denotes the certainty of the event. Therefore, an event of a higher probability would be more likely to occur. Suppose A is a particular defined event, then the expression of its probability of occurrence is as P(A). Keep on reading to learn the basics of elementary probability and its various aspects.

What is Elementary? 

What is elementary must first be dealt with before we go further into its more complex aspects. An elementary event involves only a single outcome in a space known as the sample space. Experts call such an event the sample point or the atomic event. The probability that revolves around the elementary events is known as elementary probability.

Addition Rule

The addition rule in elementary probability takes place to determine the probability of at least one of two (or more) events taking place. Generally speaking, the expression of the probability of either event A or B is as follows:

P(A or B) = P(A) + P(B) – P(A and B)

When we say that A and B are mutually exclusive, we mean that their occurrence cannot occur together, that is, P(A and B)=0. As such, when dealing with mutually exclusive events, the probability of the occurrence of either A or B can be expressed in the following manner:

P(A or B) = P(A) + P(B)

Now, consider that event A is that a person is blood group O, and event B is that the person is blood group B. These events are mutually exclusive since a person may only be one or the other. Hence, a given person is likely to be either group O or B is P(A)+P(B).

Multiplication Rule

The use of the multiplication rule in elementary probability takes place to determine the probability that two (or more) events take place together. The probability of both events A and B taking place is as:

P(A and B) = P(A) x P(B|A) = P(B) x P(A|B)

Here, the notation P(B|A) points to the probability that event B occurs when the occurrence of event A has taken place. In this case, the symbol ‘|’ actually stands for ‘given’. This is a typical case of conditional probability, in which the condition is that the occurrence of event A has taken place. For example, the probability that the ace of hearts is drawn from a well-shuffled pack is 1/51.

Bayes’ Theorem

From the multiplication rule =, the following observation can be made:

 P(A) x P(B|A) = P(B) x P(A|B)

This gives rise to the Bayes’ theorem in elementary probability, whose expression can take place as:

P(B|A)= P(A|B)P(B)/P(A)

Therefore, the B’s probability when A is given refers to the A’s probability when B is given, times the B’s probability divided by the A’s probability of A.

This formula would not be suitable if P(A)=0. This means that A’s occurrence cannot take place.

Consider an example in which event A takes place when the exercise test is positive. Also, in this example, event B takes place when angiography is positive. The expression of the probability of having both a positive exercise test and coronary artery disease is as follows:
P(T+ and D+)

Due to the application of the multiplication rule, we get:

P(T+ and D+) = P(T+|D+)P(D+)

Likelihood Ratio: 

In terms of odds, the summarization of Bayes’ theorem can take place by using the positive likelihood ratio (LR+). Its expression can take place as:

LR+ = P(T+|D+)/P(T+|D−) = Sensitivity/1−Specificity

Conclusion

Elementary probability gives the possibility of the occurrence of a defined event. Experts quantify it as a positive number between 0 and 1. Here, 0 denotes the impossibility of the event, while 1 denotes the certainty of the event. The elementary event involves only a single outcome in sample space.

Experts also call an elementary event the sample point or the atomic event. Learn the various aspects associated with an elementary probability like addition rule, multiplication rule, and the Bayes’ theorem for a good understanding of the topic.