Probability

 Definition 

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e., how likely they are going to happen, using it. Probability can range from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event. . The probability of all the events in a sample space adds up to 1.

Formula for Probability

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes

Probability of event to happen P(E) = Number of favourable outcomes/Total Number of outcomes .

Types of Probability

There are three major types of probabilities:

  • Theoretical Probability
  • Experimental Probability
  • Axiomatic Probability

Theoretical Probability

It is based on the possible chances of something to happen. The theoretical probability is mainly based on the reasoning behind probability. For example, if a coin is tossed, the theoretical probability of getting a head will be ½.

Experimental Probability

It is based on the basis of the observations of an experiment. The experimental probability can be calculated based on the number of possible outcomes by the total number of trials. For example, if a coin is tossed 10 times and head is recorded 6 times then, the experimental probability for heads is 6/10 or, 3/5.

Axiomatic Probability

In axiomatic probability, a set of rules or axioms are set which applies to all types. These axioms are set by Kolmogorov and are known as Kolmogorov’s three axioms. With the axiomatic approach to probability, the chances of occurrence or non-occurrence of the events can be quantified. The axiomatic probability lesson covers this concept in detail with Kolmogorov’s three rules (axioms) along with various examples.

Conditional Probability is the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome.

Basic Terminology :

Random Experiment :

      Suppose an experiment having more than one outcome. All possible results are knows  but the actual result connot be predicted such an experiment is called a random experiment.

Outcome :

     A possible result of random experiment is called a possible outcome of the experiment.

Sample space :

     The set of all possible outcome of a random experiment is called the sample space . The sample space is denoted by S or Greek letter omega (Ω) . The number of elements in S is denoted by n(S). A possible outcome is also callled a sample point since it is an element in the sample space.

Event:   A subset of the sample space is called an event.

Favourable Outcome :

    An outcome that belongs to the specified event is called a favourable outcome.

Types Of Events :

 Elementary Event :   An event consisting of a single outcome is called an elementary event.

Certain Event :  The sample space is called the certain event if all possible outcome are favourable outcoms i.e. the event consist of the whole sample space.

Impossible Event :  The empty set is called impossible event as no possible outcome is favourable .

Algebra Of Events : 

    Events are subset of the sample space. Algebra of events uses operation in set theory to define new events in terms of known events.

 Union of Two Events :

   Let A and B be two events on the sample space S. The union of A and B is denoted by AሀB and is the set of all possible outcomes that belong to at least one of A and B.

Ex.,   Let S = set of all positive integers not exceeding 50

      Events A =  set of all elements of S that are divisible by 6:  and 

     Event B  = set of elements of S thst are divisible by 9. find AሀB .

Solution :  A = {6, 12, 18, 24, 30, 36, 42, 48,}

                 B = {9, 18, 27, 36, 45}

∴  AሀB  = {6, 9, 12, 1, 24, 27, 30, 36, 42, 45, 48}  is the set of all elements of S that are divisible by 6 or 9.

Exhaustive Events :  Two events A and B in the sample space S are said to be exhaustive if AሀB = S.

ex.,  Consider the experiment of throwing a die and nothing the number on the top.

  Let S be the sample space

∴  S = {1, 2, 3, 4, 5, 6}

Let , A be the event that this number does not exceed 4, and  B be the event that this number is not smaller than 3.

   Then A = {1, 2, 3, 4} , B ={3, 4, 5, 6}  and therefor AሀB = {1, 2, 3, 4, 5, 6} = S 

   ∴   Events A and B are exhaustive.

Intersection of Two Events :

   Let A and B be the two events in the sample space S. The intersection of A and B is the event consisting of outcomes that belong to both the events A and B

ex.,   Let S = set of all positive integers not exceeding 50,

   Event A = set of element of S that are divisible by 3, and 

   Event B = set of all elements of S that are divisible by 5.

Then A = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48},

    B = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50}

∴  A⋂B = {15, 30, 45} is the of elements of S that are divisible by both 3 and 5.

Mutually Exclusive Events :

    Event  A and B in the sample space S are said to be mutually exclusive if they have no outcomes in commen . In other words, the intersection of mutually exclusive events is empty. MUtually exclusive events are also called disjoint events.

ex.,   Let S = set of all positive integers not exceeding 50,

   Event A =  set of elements of S that are divisible by 8, and

 Event B = set of element os S that are divisible by 13.

Then A = {8, 16, 24, 32, 40, 48}  and B = {13, 36, 39}

∴  A⋂B = ф  because no element of S is divisible by both 8 and 13.

Note :  If two events A and B are mutually exclusive, then they are called complementary events. symbolically , A and B arecomplementary events if   AሀB = S and  A⋂B = ф .


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