COMPOUND EVENTS

Probability is a branch of mathematics that deals with numerical explanations of the chances of something happening or the accuracy of a statement. In general, the probability of an event is a number between 0 and 1, with 0 signifying impossibility and 1 indicating certainty. The greater the probability of something happening, the more likely it will happen.

The word probability comes from the Latin word probabilities, which can imply “probity,” a measure of a witness’s authority in a court proceeding in Europe that is commonly linked to the aristocracy. In some ways, this differs significantly from the current definition of probability, which is a measurement of the weight of empirical evidence derived through inductive approach and statistical inference.

Types of Probability

In terms of finding the probability of an event occurring, there can be different perspectives or types of probabilities based on the nature of the outcome or the method followed. They are as follows:

• Classical Probability
• Experimental Probability
• Subjective Probability
• Axiomatic Probability

Compound event

An event (E) is a subset of the sample space (S). An event can only take place if the outcome obtained at the end of the experiment also appears to be an element of the set comprised under event E. Simple events are events comprising a single point of the sample space. If S = {29, 42, 58, 61, 74} and E = {58}, then E is a simple event. In contrast to simple events, compound events consist of more than one point of the sample space. If we take the same example of S = {29, 42, 58, 61, 74} with Event 1 = {29, 58} and Event 2 = {42, 58, 61}.

Then Event 1 and Event 2 are the compound events under-sample space S.

THEORETICAL CONCEPT OF COMPOUND EVENT: 

If any event has more than one sample point, such events are known as compound events. For example, in the experiment of “throwing a dice thrice”, the events:

E = ‘Exactly one six appeared.’

F = ‘At Least one four appeared’

G = ‘At Most one three appeared’ etc., are compound events.

all of the above events possess higher than one sample, so we can call them compound events.

   Or A’ = S – A

ODDS IN FAVOUR AND ODDS AGAINST OF A COMPOUND EVENT:

If ‘m’ outcomes are in favour of an event ‘A’ and ‘n’ outcomes are against of an event ‘A’, then we can say:

Odds in favour of event A =

mn=no of outcomes which are in favour of event Ano of outcomes which are not in favour of event A

Odds in against of event A = 

nm=no of outcomes which are not in favour of event Ano of outcomes which are in favour of event A

NOTE: If PA=ab, then

(i) Odds in favor of compound event A = a:b-a

(ii) Odds in against of compound event A = b-a:a

Conditional Probability Distribution 

The conditional probability distribution of Y/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 considering 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.

CONCLUSION:

In this article, constituting of study material notes on compound events, we learned about the compound events and algebra of events using the concepts of compound events like – The event ‘A or B’, The event ‘A and B’, the event ‘neither A nor B’ and the event ‘A but not B’ along with an illustrated example. With the help of compound events study material, we also read about odds in favour and odds against compound events. To gain a better understanding of this chapter other topics such as the addition theorem of probability, students should read conditional probability and multiplication theorem on probability .