Probability

    Probability

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur or how likely it is that a proposition is true. Probability is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility and 1 indicates certainty. 

example : It will probably rain today.

              :It doubt that he will pass  the test.

           Types of  probability : 

1. An Experimental approach

2. Axomatic approach

Introduction :-

• Event

• Sample Space

• Exhaustive cases

Event :- Let E be the event of any Experiment, and their outcome be probably defined, then total no. of Event is denoted by n(E).

Sample Space :- It is collection of all event.It is denoted by 's'.

                          let E1,E2,------------,En be the event.             then             S={E1,E2,E3,-------------,En}            n(s) = n.Probability Event :- Let E be the event and s be the sample space then the probability is defined as p(E)
                     total number of events   p(E)   =  ---------------------------------                     total  number of outcome         i.e   p(E) = n(E) / n(S)where 0<=p(E)<=1
Q. When one coin is tossed what is probability of getting tail .     let E be the event           S =  {H,T}  n(S) = 2
           E = {T}  n(E) = 1
       p(E) = n(E) / n(S)   1 / 2

Q.    what is probability of getting head.

             let E be the event  

           S =  {H,T}  n(S) = 2
           E = {H}  n(E) = 1
       p(E) = n(E) / n(S)   1 / 2

Dependent and independent events

Independent events

Two events, A and B are independent if and only if

     P(A and B)=P(A)×P(B)

Independence

Roll a single 6$\text{6}$-sided die and consider the following two events:

• E: you get an even number
• T: you get a number that is divisible by three

Now answer the following questions:

• What is the probability of E?
• What is the probability of getting an even number if you are told that the number was also divisible by three?
• Does knowing that the number was divisible by 3 change the probability that the number was even?

Are the events E and T dependent or independent according to the definition? (Hint: compute the probabilities in the definition of independence.)

The probability of A is the ratio between the number of outcomes in A and the number of outcomes in the sample space, S.

P(A)=n(A)n(S)

Now, let's say that we know that event B happened. How does this affect the probability of A? Here is how the Venn diagram changes:

A lot of the possible outcomes (all of the outcomes outside B) are now out of the picture, because we know that they did not happen. Now the probability of A happening, given that we know that B happened, is the ratio between the size of the region where A  is present (A and B) and the size of all possible events (B).

P(A if we know B)=n(A and B)n(B)