Define Experiment in Probability :
The term experiment mean a plan or activity or process whose result yield a set of data.
A single performance of an experiment is called trial.
The result obtain from an experiment or trial is called outcome.
Random Experiment in Probability :
An experiment which produces different results it is repeated a large number of times under the same condition is called random experiment.
Examples :
- The tesing of coin.
- The throwing of balance die.
Random experiment has three properties:
- The experiment can be repeated practically and theoretically any number of time.
- The experiment always has two or more possible outcomes.
- The outcome are result of each repetition is unpredecable that is. It has some degree of unstantity.
Define the sample space of a random experiment:
A set consisting of all possible outcome that can result from random experiment is defined to be sample space for the experiment. It is denoted by letter capital S.
Each possible outcome is a member of simple space and is called simple coin.
The sample space for this experiment may be express in set notation as S={H, T} sample space for testing two coins will contain four possible outcome denoted by
S={ HH, HT, TH, TT }
What are Events in statistics :
An event is an individual outcome or any number of outcome of random experiment. Any subset of sample space S of the experiment is called an event.
An event that contains exactly one sample point is define a sample point.
A compound event contain more than one sample point and it produced by the union of sample event.
Mutually Exclusive Events or Disjoint Event
Two events A and B of a single experiment are said to be mutually exclusive or disjoint. If and only if they cannot both occur in the same that is they have no point in common.
For example, when we toss a coin we get either head or tail but not both. The two event head and tail are mutually exclusive event.
Exhaustive event
Events are said to be collectively exhaustive when the union of mutually exclusive event is the entire sample space.
For example, In our coin tossing experiment, head and tail are collectively exhaustive set of events.
Equally likely Events :
In other words each event should occur in equal number in repeated trial.
For example, when a fair coin is toss the head is as likely is as to appear as the tail and the proportion of times each side is expected to appear in one by two.
Counting Sample Points :
When the number of sample event in a sample space 'S' is very large it becomes very inconvenient and difficult to list them all and to count the number of point in the sample space 'S' and in subset of sample space. We then need some methods and rules which help us to count the number of all sample points. A few of the basic rules or methods use in accounting as given below :
- Rule of multiplication
- Rule of permutation
- Rule of combination
Probability of an Event :
If an event 'A' is defined in a sample space 'S' then its probability will be equal to the sum of the probabilities of all sample points that are included in event 'A'. When all the 'n' possible of random experiment are equally likely to occur the each out is assigned the same probability one over n (1/n).
Conditional Probability :
The sample space of an experiment must be change when some addition information to the outcome of the experiment. The effect of such information is to reduce the sample space 'S' by excluding some outcomes has been impossible which be far is receiving the information.
The probability associated with such a reduce sample space are called conditional probability.
Independent Events :
Two events A and B in the sample space 'S' define to be independent if the probability that one event occurs is not affected by whether the event as a has not occurs.
Dependent Event :
The event A and B are defined to be dependent if
P(A ∪ B) ≠P(A) . P(B)
This mean that the occurrence of one of the event in some way affected the probability of the occurrence of the other event.
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Random variable :
Usually we are not interested in particular outcome of a random experiment but our interest relate to some numerical description of the outcome.
For example, we two coins are toss. We may be interested only in number of head which appear and not in the actual sequence of head and tail which is not a numerical quantity. To express the outcome is number we assign to each non-numerical outcome of the sample space S={HH, HT, TH, TT}.
Discrete probability distribution :
An arrangement of all possible values of a random variable along with their respective probabilities is called discrete probability distribution.
Discrete Random Variable :
A random variable which can take only some specific and finite set a value in a given range and these values cannot be further subdivided are called discrete random variable .
Examples :
- Number of accident per day.
- Number of students in university.
Continues Random Variable :
A random variable which can take every possible value in the given range and these value can take decimal points. Continues random variable are obtain by some measurements.
For example :
- Height of student.
- Weight of student.
- Temperature
Probability Density Function :
The probability distribution of continuous random variable is called continuous probability distribution or probability density function. It provides a formula or function with the help of which we can find the probability of any interval in a given range.
P(a<x<b) = ₐ∫ᵇ f(x) dx
Problem :
Find also probability that the sample value will exceed 1.
Problem 2 :
Find the P.D of the sum of dot when two fair dice are thrown.
Use probability distribution to find the probability of obtaining.
