Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 6 Probability will enable students to study smartly.

## CHSE Odisha 12th Class Math Notes Chapter 6 Probability

Important terms:

(a) Random experiment: It is an experiment whose results are unpredictable.

(b) Sample space: It is the set of all possible outcomes of an experiment. We denote the sample space by ‘S’.

(c) Sample point: Each element of a sample space is a sample point.

(d) Event: Any subset of a sample space is an event.

(e) Simple event: It is an event with a single sample point.

(f) Compound event: Compound events are the events containing more than one sample point.

(g) Mutually exclusive events: Two events A and B are mutually exclusive if A ∩ B = φ (i.e. occurrence of one excludes the occurrence of other)

(h) Mutually exhaustive events: The events A_{1}, A_{2}, A_{3} ……. A_{n} are mutually exhaustive if A_{1} ∪ A_{2} ∪ A_{3} ∪ A_{n} = S.

(i) Mutually exclusive and exhaustive events.

The events A_{1}, A_{2}, A_{3} ……. A_{n} are mutually exclusive and exhaustive if

(i) A_{1} ∩ A_{2} = φ for i ≠ j

(ii) A_{1} ∪ A_{2} ∪ …… A_{n} = S.

(j) Equally likely events: Two events are equally likely if they have equal chance of occurrence.

(k) Impossible and certain events: φ is the impossible and S is the sure or certain event.

(i) Independent events: The events are said to be independent if the occurrence or non-occurrence of one does not affect the occurrence of non-occurrence of other.

Probability of an event:

Let S be the sample space and A is an event then the probability of A is

\(P(A)=\frac{|A|}{|S|}=\frac{\text { No.of out comes favourable to } A}{\text { Total number of possible outcomes. }}\)

Note:

1. P(φ) = 0

2. P(S) = 1

3. P(A’) = P (not A) = 1 – P(A)

4. P(A) + P(A’) = 1

Odds in favour and odds against an event:

Let in an experiment is the number of cases favourable to A and ‘n’ is the number of cases not in favour of A then

Addition theorem:

If A and B are any two events then

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Note:

If A and B are mutually exclusive then

P(A ∪ B) = P(A) + P(B)

(∴ P(A ∩ B) = P(φ) – 0)

Conditional probability:

Let A and B are any two events and P(B) ≠ 0 then the conditional probability of A when B has already happened

P(A/B) = \( \frac{P(A \cap B)}{P(B)} \)

Note:

1. P(A ∩ B) = P(B) . P(A/B)

2. If A and B are mutually independent events then P(A/B) = P(A).

∴ P(A ∩ B) = P(A) . P(B)

3. If A and B are independent events then (i) A’ and B’ (ii) A’ and B (iii) A and B’ are also independent.

4. P(A_{1} ∩ A_{2} ∩ A_{3} ….. ∩ A_{n}) = P(A_{1}) . P(A_{2}/A_{1}) . P(A_{3}/A_{2} ∩ A_{1}) ….. P(A_{n}/A_{1} ∩ A_{2} ∩ …. ∩ A_{n-1})

5. Let A_{1}, A_{2} ….. A_{n} are mutually exhaustive and exclusive events and A is any event which occurs with A_{1} or A_{2} or A_{3} … or A_{n} then

P(A) = P(A_{1}) . P(A/A_{1}) + P(A_{2}) . P(A/ A_{2}) + …….. + P(A_{n}) . P(A/A_{n}).

This is called the total conditional probability theorem.

Baye’s theorem:

If A_{1}, A_{2} …… A_{n} are mutually exclusive and exhaustive events and A is any event which occurs with A_{1} or A_{2} or A_{3} or …. A_{n} then

\( P\left(A_i / B\right)=\frac{P\left(A_i\right) \cdot P\left(A / A_i\right)}{\sum_{i=1}^n P\left(A_i\right) P\left(A / A_i\right)} \)