Introduction to study probability and statistics:

Probability is a way of expressing a knowledge or belief that the event will occur. In mathematics the concept has given a exact meaning in study of probability and stastics theory, that is used extensively in such areas of study in mathematics, studying statistics,finance,gambling, science, and philosophy to draw conclusion about the potential events and the underlying mechanics of the complex systems.

Statistics is the science of making some effective use of numerical data relating to all groups of individuals. It deals with all aspects of the analysis and interpretation of such data, but also the planning of the collection of data, in terms of design of the studying surveys and experiments.

  

Overview of study probability and statistics:

 

  • Population parameters are typically denoted by lower-case Greek letters (e.g:μ,σ,π, etc.)
  • Random variables are always denoted by capital letters (e.g:X, Y, U, W, etc.) and specific measurements are denoted by lower-case letters (e.g: y, u, w, etc.).
  • Data driven estimates for specific population parameters that are always denoted by an corresponding symbol with an over-line.
  •  p(AuB) indicates the probability that the events A and B both occur.
  •  p(AnB)indicates the probability of either an event A or event B occurring ("or" in this case means one or the other or both).

 

Notations of Study Probability and statistics:

 

  • P(A) refers to the probability that event of A will occur.
  • P(A|B) refers to the conditional probability that event of A occurs, given that event of B has occurred.
  • P(A ∩ B) refers to the probability of the intersection of events A and the event B.
  • h(x; N, n, k) refers to the hypergeometric probability.
  • g(x; P) refers to the geometric probability.
  • b(x; n, P) refers to the binomial probability.
  • E(X) refers to the expected value of the random variable.

 

Examples:

Bowl 1contains 6 red chips and 4 blue chips. Five of these 10 chips are selected at random and without replacement and put in bowl 2 which was originally empty. One chip is then drawn at random from bowl 2 Given that this chip is blue, find the conditional probability that 2 red chips and 3 blue chips are transferred from bowl 1to bowl 2.

Solution:

      
P(3 B taken from bowl 1 and B chip drawn from bowl 2)=(6c2*4c3)/10c5*3/5

                                                         P(B chip drawn from bowl 2)= (6c4*4c1/10c5)1/5+(6c3*4c2)/10c5*2/5+          

                                                                                                                (6c2*4c3)/10c5*3/5+(6c1*4c4)/10c5*4/5

                                                                                                           =5/14