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# Probability

## Probability Synopsis

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, ... En 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

Top Formulae
1. 0 ≤ P(B|A) ≤ 1

2. If E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E given that F has occurred, i.e. P(E|F) is given by

3. Multiplication theorem
a. For two events:
If E and F are two events associated with a sample space S, then
= P(E) P(F|E) = P(F) P(E|F) provided P(E) ≠ 0 and
P(F) ≠ 0.
b. For three events:
If E, F and G are three events of sample space S, then

4. Multiplication theorem for independent events

5. Let E and F be two events associated with the same random experiment. The two events E and F are said to be independent if
i. P(F|E) = P(F) provided P(E) ≠ 0 and
ii. P(E|F) = P(E) provided P(F) ≠ 0
iii.  = P(E).P(F)
6. Theorem of total probability
Let {E1, E2,...,En} be a partition of the sample space S, and suppose that each of the events E1, E2, ... En has non-zero probability of occurrence. Let A be any event associated with S, then
P(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + ... +  P(En) P(A|En)