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.
→ 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
→ 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