"Something likely to happen !!"
1. Classical (Priori) Definition of Probability:
If $n =$ Exhaustive, Mutually exclusive and equally likely cases, and
$m =$ Favorable cases to an event $(E)$
Then Probability of the happening of an event $(E)$ denoted by $P(E)$ is defined by,
$P(E) = \frac{m}{n}$ ................... (i)
The Probability $P(E)$ of happening of an event $E$ satisfies the following property:
$0 \leq P(E) \leq 1$
Case 1. If $E$ is an impossible event then $P(E) = 0$If $E$ is an sure event then $P(E) = 1$
Case 2: The sum of the probabilities of the occurrence $P(E)$, and non-occurrence $P(\bar{E})$ of an event is unity.
i.e. $P(E) + P(\bar{E}) = 1$
2. Probability:
If $S$ be the sample space of random experiment.$n(S)$ = number of sample points of random experiment$n(E)$ = number of favourable points to an events (E), Then,
The Probability of happening an event $P(E)$ is defined by:
$P(E) = \frac{n(E)}{n(S)}$ ............................ (ii)
3. Two basic laws of Probability:
i) Additional theorem:
If $A$ and $B$ are two events with their respective Probabilities $P(A)$ & $P(B)$.
» Then the probability of occurrence of at least one of these two events denoted by $P(A \cup B)$ is given by:
$P(A$ or $B)$ = $P(A \cup B)$ = $P(A) + P(B) - P(A \cap B)$ .......(iii)
where $P(A \cap B)$ is the probability of the simultaneous occurrence of the events $A$ and $B$ (i.e common to $A$ and $B$).
ii) Multiplication theorem:
If two events $A$ and $B$ are independent, then the probability of their simultaneous occurrence is equal to product of their individual probabilities.
$P(A$ and $B)$ = $P(A \cap B)$ = $P(A) . P(B) $ ...........(iv)
4. Permutation and Combination:
i) Permutation: The arrangement of objects in some order.If $n$ = total number of objects (number of permutation)$r$ = number of object taken (number of ways)
» The number of permutations of a set of $(n)$, different objects taken $(r)$ at a time denoted by $P(n,r)$ or $^{n}\textrm{P}_r$ is:
$^{n}\textrm{P}_r$ = $P(n,r)$ = $\frac{n!}{(n-r)!}; (r\leq n)$ .......................... (v)
Where $n!$ = factorial $n = 1,2,3, .............. n.$Also,
$P(n,n)$ = $^{n}\textrm{P}_n$ = $n!$; $(0! = 1)$
» The number of permutations of a set of $n$ objects taken all of them at a time where $p$ of them are of one kind, $q$ of them the second kind, $r$ of them of the third kind. Then,
Total number of permutation = $\frac{n!}{p!q!r!}$ .......................(vi)
ii) Combination: The selection of objects without regard to any order of arrangement.If $n$ = total number of selection (combination)
$r$ = different objects taken at a time.
» The total number of combinations of $n$ objects taken $r$ at a time,
$C(n,r)$ = $\frac{n!}{(n-r)!r!}$ ................(vii)
5. Binomial Distribution:
An experiment consisting only two outcomes (Success and failure) is known as "Bernoully Process". The discrete probability distribution derived from the Bernoully process is known as Binomial distribution.
Let $p$ be the probability of a success and $q$ be the probability of a failure in one trial. Let $r$ be the number of success described in $n$ independent trials of the binomial expansion of $(p+q)^n$, Then there would be $(n-r)$ failures.
The probability of getting exactly $r$ success and consequently $(n-r)$ failure in $n$ independent trials is given by;
$P(r)$ = $^{n}\textrm{C}_r \;p^r \;q^{n-r}$; $(0\leq r\leq n)$ ............... (viii)
Where, $P(r)$ = Probability of $r$ successive in $n$ trials,
$n$ = number of trial performed,
$p$ = probability of a success in a trial,
$q$ = probability of a failure in a trial such that $p + q = 1$
$r$ = number of successive in $n$ trials. (r = 0, 1, 2, ........n).
Mean of binomial distribution = $np$ ........................ (ix)
Standard deviation of binomial distribution = $\sqrt{n\;p\;q}$ ..................(x)
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