# Probability Theory

**Experiments:**

An experiment repeated under essentially homogeneous and similar conditions results in an outcome, which is unique or not unique but may be one of the several possible outcomes. When the result is unique then the experiment is called a deterministic experiment. Any experiment whose outcome cannot be predicted in advance, but is one of the set of possible outcomes, is called a random experiment. If any experiment as being performed repeatedly, each repetition is called a trial. We observe an outcome for each trial. If a coin is tossed, the result of a particular tossing, i.e., whether it will fall with head or tail can not be predicted. A die with faces marked with numbers numbers 1,2,3,4,5,6. When we throw a die the number on upper face, we can not predict which face will be on the top.

**Sample Space:**

The set of all possible outcomes of a random experiment is called the Sample Space. e.g., The sample space for tossing coins contains just two outcomes - Head or Tail. If we denote outcome of head as H and of tail as T , The sample space for the random experiment of tossing of coins is S = { H,T }

Similarly, the sample space of throwing of a die will be S = { 1,2,3,4,5,6 }

Each possible outcome given in the sample space is called an element or sample point. Here in case of die, the sample points are 1,2,3,4,5,6. Further sample space may be classified in two way based on number of sample points are finite or infinite. A sample space is called discrete if it contains finite or finitely many countably infinite sample points. for example-when a coin and die is thrown simultaneously, the sample space is given below is discrete:

S = { (H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6) }

In other example, a coin is tossed until a head turns up. The sample space is S = { H,TH,TTH,TTTH, ...}

where the number of sample points is infinite but there is one to one correspondence between the sample points as natural numbers, i.e. S is countable infinite. Hence is a discrete sample space. On the other hand a sample space is said to be continuous if it contains uncountable number of sample points. e.g., All points on a straight line, All points inside a circle of radius 4, etc.

**Events:**

Any subset of the sample space S, associated with any random experiment is called an event. For example when a die is thrown, then the sample space is

S = 1,2,3,4,5,6 and its subset E1 = { 2,4,6 }. Represents an event, which represents ‘The number on upper face is even’. Since a set is also a subset of itself, we say that sample space is itself an event and call it a sure event. An event that contains no sample point is called an impossible event. An event containing exactly on sample point is called simple or elementary event. During the performance of an experiment, those point which entail or assume the occurrence of an event A, are called favorable to that event. If any of the sample point favorable to an event A occurs in a trial, we say that the events occurs at this trial.

**Complex or Composite Event:**

The union of simple events is called composite event. In other words if an event can be decomposed into simple events, then it is called a composite event. Let A and B be two simple events associated with sample space S, then the event which consists of all the sample points which belong to A or B or both, is called the union of A and B as A ∪ B or A B. No sample point is taken twice in A ∪ B. For example, when three coins are tossed together, then the event E = at least two heads, is a composite event.

S = { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT }

A(Event of two heads) = { HHT, HTH, THH }

B(Event of Three heads) = { HHH }

E = A∪B = { HHH, HHT, HTH, THH }

**Compound or Joint Event:**

The occurrence of two or more events together is called a compound event. In other words the intersection of two or more events is called compound event. Let A and B be any two simple events associated with a sample space S, then the intersection of A and B, written as A ∩ B or AB, which contains all those point of S, which are common in A and B. e.g.,

S = { 1,2,3,4,5,6 }

A(Event of Even points) = { 2,4,6 }

B(Event of multiple of Three) = { 3,6 }

E = A∩B = {6}

**Mutually Exclusive or Disjoint Events:**

Two events are called mutually exclusive if they can not occur simultaneously. In other words two or more events are said to be mutually exclusive if the occurrence of any one of the event precludes the occurrence of the other events. There are no sample points are common in mutually exclusive events. For example, if we toss a coin once, the events of head and tail are mutually exclusive because both can not occur same time.