Summary of Class 12 Maths Probability
Conditional
Probability
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
P(E/F) = 
, where p(F) ≠ 0
Conditional probability is the probability
that an event will occur given that another event has already occurred.
Properties of Conditional
Probability
Let E and F be events of a sample space S of an experiment, then
we have
1. P(S|F) = P(F|F) = 1
2. P((A 
B)|F) = P(A|F) + P(B|F) – P((A 
B)|F)
Where A and B are any two events associated with S.
3. P(E’|F) = 1 − P(E|F)
Multiplication Theorem on Probability
1. Let E and F be two events associated with a random experiment,
then
P(E 
) = P(E) . P(F/E), P(E) ≠ 0
= P(F) . P(E/F), P(F) ≠
0
2. Let E, F and G be three events
associated with a sample space S, then
P(E 
) = P(E) . P(F/E) . P(G/E )
Independent
Events
Two events E and F are said to be
independent, if the occurrence or non-occurrence of one event does not affect
the occurrence or non-occurrence of another event.
Thus, two events E and F will be
independent, if
a. P(F/E) = P(F), P(E) ≠ 0
b. P(E/F) = P(E), P(F) ≠ 0
Let E and F be two events associated with the same random
experiment, then E and F are said to be independent, if
P(E ) = P(E). P(F)
Partition of a Sample Space
A set of events E1, E2, ..., En
is said to represent a partition of the sample space S if
(a) Ei Ej
= φ, i ≠ j, i, j = 1, 2, 3, ..., n
(b) E1 E2
... En
= S and
(c) P(Ei) > 0 for all i = 1, 2, ..., n.
Theorems of
Total Probability
Let {E1, E2, ..., En} be a partition of the sample space S. 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)
= 
Bayes’ Theorem
Let E1, E2
,..., En are n non-empty events which constitute a
partition of sample space S, i.e. E1, E2 ,..., En
are pairwise disjoint and E1 E2 ... En = S. Let A is any event
of non-zero probability, then
where k = 1, 2, 3, …, n
Random
Variable
A random variable is a real valued function whose domain is the sample space of a random experiment. It is generally represented by the letter X.
Probability
Distribution of a Random Variable
Let X
is a random variable and takes the values x1, x2, x3,
…, xn with respective probabilities p1, p2, p3,
…, pn. Then, the probability distribution of X is represented by
|
X |
x1 |
x2 |
x3 |
… |
xn |
|
P(X) |
p1 |
p2 |
p3 |
… |
pn |
Where pi > 0 such that 
;
i = 1, 2, 3, …, n
Mean
of a Random Variable
Let X
is a random variable and takes the values x1, x2, x3,
…, xn with respective probabilities p1, p2, p3,
…, pn.
Mean
of random variable X, denoted by μ [or expected value of X denoted by E(X)] is
defined as
μ = E(X) = 