Probability

Synopsis

1. A random experiment is an experiment which produces a set of well-defined possible outcomes under identical conditions.
   
   
2. Elementary event: Each outcome of the random experiment is called an elementary event.
   
   
3. Sample space: The set of all possible outcomes of the random experiment is called the sample space.
   
   
4. Event: In a random experiment, the subset of the sample space is called an event.
   
   
5. Certain event (sure event): If a random experiment occurs always, then the corresponding event is called a certain event.
   
   
6. Impossible event: If a random experiment never occurs, then the corresponding event is called an impossible event.
   
   
7. Mutually exclusive event: In a random experiment, if the occurrence of any one of the event prevents the occurrence of all the other events, then the corresponding events are said to be mutually exclusive.
   
   
8. Exhaustive event: In a random experiment, if the union of two or more events forms the sample space, then the associated events are said to be exhaustive events.
   
   
9. Probability of an event: In a random experiment of n elementary events, if m events are favourable to an event A, then the probability of occurrence of A is denoted by P(A) and P(A) = .
   
   
10. Probability of a sure event is 1 and probability of an impossible event is 0.
    
    
11. Addition theorem: In a random experiment, if A and B are two associated events, then
    
    
    
12. If A and B are two mutually exclusive events, then
    
    
    
13. In a random experiment, if A, B and C are three associated events, then
    
    
    
14. If A, B and C are three mutually exclusive events, then
    
    
    
15. If A and B are two events, then the probability of occurrence of A only is
    
    
    
16. If A and B are two events, then the probability of occurrence of B only is
    
    
    
17. If A and B are two events, then the probability of occurrence of exactly one of A and B is 18.
    
    
    
18. The probability that event B will occur, given the knowledge that event A has already occurred, is called conditional probability. It is denoted as P(B|A).
    
    
19. Conditional probability of B given A has occurred P(B|A) is given by the ratio of the number of events favourable to both A and B to the number of events favourable to A.
    
    
    
20. If E and F are two events associated with sample space S, then 0 ≤ P (E/F) ≤ 1.
    
    
21. If E and F are the events of a sample space S of an experiment, then
    i. P(S|F) = P(F|F) = 1
    ii. For any two events A and B of sample space S, if F is another event such that P(F) ≠ 0
    
    iii. P(E’|F) = 1 − P(E|F)
    
    
22. Multiplication rule of probability for more than two dependent events, If A, B and C are three events of sample space, we have
    
    
    
23. Two events A and B are independent if and only if the occurrence of A does not depend on the occurrence of B and vice versa.
    
    
24. If events A and B are independent, then P(B|A) = P(B) and P(A|B) = P(A).
    
    
25. Three events A, B and C are independent if they are pairwise independent i.e.   
    
    
    
26. Three events A, B and C are independent if
    
    Independence implies pairwise independence, but not conversely.
    
    
27. Bayes' theorem is also known as the formula for the probability of ‘causes’.
    
    
28. If E₁, E₂, ... E_n are n non-empty events which constitute a partition of sample space S and A is any event of non-zero probability, then by Bayes’ theorem