Learning these TS Inter 2nd Year Maths 2A Formulas Chapter 9 Probability will help students to solve mathematical problems quickly.

## TS Inter 2nd Year Maths 2A Probability Formulas

→ An experiment that can be repeated any number of times under essentially identical conditions in which

- All possible outcomes of the experiment are known in advance.
- The actual outcome in a particular case is not known in advance, is called a “Random experiment”,

E.g :

- In tossing an unbiased coin, we have only two possible outcomes: Head (H) and Tail (T). We can find the outcome in only one particular trial and the experiment can be done any number of times under essentially identical conditions.
- Rolling a die; denoting six faces of a cubical die with the numbers 1, 2, 3. 4. 5 and 6, the possible outcome of the experiment is one of the numbers 1, 2, 3, 4, 5 or 6.

→ Any possible outcome of a random is called an “Elementary event” or “Simple event” denoted by ’E’.

- The set of all elementary events of a random experiment is called the sample space S associated with the event ’E’.
- An elementary event is a point of the sample space S.
- A subset E of S is called an event. That is a set of elementary events is called an event.
- The complement of the event E denoted by E
^{c}is the event given by E^{c}= S – E which is called the complementary event of E. - The set Φ and the set S are trivial subsets of S are events called impossible event and certain event.

→ Events E_{1} E_{2}, ……………… E_{n} are said to be mutually exclusive if E_{i} ∩ E_{j} = Φ for i ≠ j, 1 ≤ i, j ≤ n.

→ Events E_{1} E_{2}, ……………… E_{n} are said to be equally likely if there is no reason to expect one of them to happen in performance to the other.

→ Events E_{1} E_{2}, ……………… E_{n} are called exhaustive events if E_{1} ∪ E_{2} ∪ ……………… ∪ E_{n} = S.

→ Classical definition of Probability : If a random experiment results in n exhaustive, mutually exclusive and equally likely ways and m out of them are favourable to the happening of an event E, then the probability of E denoted by P(E) = \(\frac{m}{n}\)

For any E, 0 ≤ P( E ) ≤ 1.

→ Relative frequency definition of Probability : Suppose E is an event of a random experiment. Let the experiment be repeated n times out of which E occurs m limes. Then the ratio \(\frac{m}{n}\) is called the nth relative frequency of the event E. If a real number ‘l’ such that \({Lim}_{n \rightarrow \infty}\) r^{n} = l then l is called the Probability of E.

→ Kolmogorov’s Axiomatic definition of Probability :

Suppose S is the sample space of the random experiment and S is finite. Tiien a function P : P(S) → R satisfying the following axioms is called the Probability function.

- P(E) ≥ 0 ∀ E ∈ P( S )
- P(S) = 1
- If E
_{1}, E_{2}∈ P( S ); then P( E ) is called the Probability of E. If S is countable of infinite then the (iii) axiom is \(P\left(\bigcup_{n=1}^{\infty} E_n\right)=\sum_{n=1}^{\infty} P\left(E_n\right)\)

→ P(Φ) = 0 and P(S ) = 1.

→ Addition theorem on Probability : If E_{1} and E_{2} be any two events of a random experiment, then

P(E_{1} ∪ E_{2}) = P(E_{1}) – P( E_{2}) – P(E_{1} ∩ E_{2})

→ Conditional Probability of the occurrence of an event A, given that B has already happened is denoted by P(\(\frac{A}{B}\)) and is defined as P\(\left(\frac{A}{B}\right)=\frac{P(A \cap B)}{P(B)}\), P(B) ≠ 0

→ Multiplication theorem on Probability: If A and B are events of a sample space S and P (A) > 0, P (B) > 0 then P (A ∩ B) = P(A). P(\(\frac{B}{A}\)) = P (B) . P(\(\frac{A}{B}\))

→ Independent events: Events A and B of a sample space S are said to be independent if P(A ∩ B) = P(A) – P(B) otherwise they are said to be dependent.

→ Baye’s theorem : If E_{1}, E_{2}, ………….. E_{n} are mutually exclusive and exhaustive events of a

random experiment with P(E_{i}) > 0 for i = 1, 2, …………………. n then

\(P\left(\frac{E_k}{A}\right)=\frac{P\left(E_k\right) \cdot P\left(\frac{A}{E_k}\right)}{\sum_{i=1}^n P\left(E_i\right) P\left(\frac{A}{E_i}\right)}\), k = 1, 2, …….. n