TS Inter 2nd Year Maths 2A Probability Formulas

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 Ec is the event given by Ec = 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 E1 E2, ……………… En are said to be mutually exclusive if Ei ∩ Ej = Φ for i ≠ j, 1 ≤ i, j ≤ n.

→ Events E1 E2, ……………… En are said to be equally likely if there is no reason to expect one of them to happen in performance to the other.

TS Inter 2nd Year Maths 2A Probability Formulas

→ Events E1 E2, ……………… En are called exhaustive events if E1 ∪ E2 ∪ ……………… ∪ En = 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 \({Lim}_{n \rightarrow \infty}\) rn = 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 E1, E2 ∈ 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 E1 and E2 be any two events of a random experiment, then
P(E1 ∪ E2) = P(E1) – P( E2) – P(E1 ∩ E2)

→ 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

TS Inter 2nd Year Maths 2A Probability Formulas

→ 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 E1, E2, ………….. En are mutually exclusive and exhaustive events of a
random experiment with P(Ei) > 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

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