Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

PROBABILITY Part-2

PROBABILITY PART-2

This part is the continuation of Probability part-1. This Post is specially for the students after 10th standard. For complete knowledge first students study Probability Part-1 and then part-2

Main points to be discussed here

1

Random Experiment

2

Different types of events

3

Variable description of the events

4

Conditional Probability

5

Multiplication Theorem on Probability

6

Laws of Total Probability

7

Baye's Theorem

8

Bernoulli's Trial

9

Mean and Variance of the Distribution

10

Shortcut Method of finding Mean, Variance and Standard Deviation.


Random Experiment:- 
An experiment whose outcomes cannot be predicted or determined in advance are called a random experiment.

Elementary events:- 
If a random experiment performed, then each of its outcome is known as an elementary event.

Sample space:- 
The set of all possible outcomes of a random experiment is called the sample space.

Event:- A subset of the sample space is called an event.

Simple Event :-
An event having only one sample point is called the simple event. eg:- HH, TT, HHH, 222 .... etc.
Compound event:- 
An event having more than one sample point is called a compound event. eg:- HT, HTH,  123 ... etc.

Mutually exclusive events:-
Two or more events are said to be mutually exclusive events if their intersection is a null set eg:- A and B are said to be mutually exclusive events If A∩B = Φ

Exhaustive events:-  
Two or more events are said to be exhaustive events, if their union is a sample space, For Example:-  A ∪ B ∪ C ∪ ..... = S

Conditional probability:- 
Conditional probability of two events A and B is denoted by P(A|B) and is called as
P(A|B) = Probability of occurrence  of A given that B has already occurred.

Conditional Probability is calculated by using following formula.

\[P(A|B)=\frac{P(A\cap B)}{P(B)}\]


Independent events:- If A and B are independent events then 

\[P(A\cap B)=P(A).P(B)\]

\[P(A\cap B\cap C)=P(A).P(B).P(C)\]


Explanation of Some symbols and Terms used in the problems

Variable Description of the event  Set theory notation

A or B (At least one of A or B) = A B

A and B = A ∩ B ,                    Not A = Ā

\[A \: but \: not\: B = A\cap\overline{B}\]

\[Either\; A\: or\: B=A\cup B\]

\[Neither\: A\: nor\: B=\overline{A}\cap \overline{B}\]

\[All \: three\: of\: A, B \: and \: C = A\cap B\cap C\]

\[At\: least\: one\: of\: A, B, or\: C=A\cup B\cup C\]

\[Exactly \: \: one\: of\: A\: and\: B = (A\cap \overline{B})\cup (\overline{A}\cap B)\]

\[D' Morgan's Law:-\: A'\cap B'=(A\cup B)'\]\[OR\: \: \: A'\cup B'=(A\cap B)'\]

\[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]\[P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)\]\[-P(B\cap C)-P(C\cap A)+P(A\cap B\cap C)\]

************************************

Multiplication Theorem On Probability

If A and B are two events associated with a random experiment, then

\[P(A\cap B)=P(A)P(B/A)\: where P(A)\neq 0\]

\[P(A\cap B)=P(B)P(A/B)\: where P(B)\neq 0\]

Extension of Multiplication Theorem

\[If A_{1},A_{2},A_{3},.....,A_{n}\: are\: n \: events\: associated\: with\: the\: random\: experiment,\: then\]

\[P(A_{1}\cap A_{2}\cap A_{3}.....\cap A_{n})=P(A_{1})P(A_{2}/A_{1})P(A_{3}/A_{1}\cap A_{2})P(A_{4}/A_{1}\cap A_{2}\cap A_{3}).....\]

\[........P(A_{n}/A_{1}\cap A_{2}\cap .....\cap A_{n-1})\]

The laws of total probability:- 
\[Let\: S\: \: be\: the\: sample\: space \: and \: let\: E_{1},E_{2},E_{3},....,E_{n} \: be \: n \: mutually\: exclusive\]\[and\: exhaustive\: events.\]\[If \: A \: is \: any\: event,\: then\]\[P(A)=P(E_{1})P(A/E_{1})+P(E_{2})P(A/E_{2})+.....+P(E_{n})P(A/E_{n})\]

BAYE'S THEOREM

\[Let \: S\: be\: the \: sample\: space \: and\: let\: E_{1},E_{2},E_{3},.....,E_{n}\: be\]

\[mutually \: exclusive\: and\: exhaustive\: events\: associated\: with\: a\: random\: experiment.\]

\[If\: A \: is\: any \: event\: which\: occurs\: with\: E_{1} \: or\: E_{2}\: or\: E_{3}\: or...or\: E_{n}\]

\[P(E_{i}|A)=\frac{P(E_{i})P(A|E_{i})}{\sum P(E_{i})P(A|E_{i})}\: \: i=1,2,3....,n\]

If n = 2 then Baye's Theorem written as: 

\[P(E_{1}|A) =\frac{P(E_{1})P(A|E_{1})}{P(E_{1})P(A|E_{1}) +P(E_{2}) P(A|E_{2})}\]

\[P(E_{2}|A)=\frac{P(E_{2})P(A|E_{2})}{P(E_{1})P(A|E_{1})+ P(E_{2}) P(A| E_{2})}\]

If n = 3 then Baye's Theorem written as:

\[P(E_{1}|A)=\frac{P(E_{1})P(A|E_{1})}{P(E_{1})P(A|E_{1})+P(E_{2})P(A|E_{2}) +P(E_{3})P(A|E_{3})}\]

\[P(E_{2}|A)=\frac{P(E_{2})P(A|E_{2})}{P(E_{1})P(A|E_{1})+P(E_{2})P(A|E_{2}) +P(E_{3})P(A|E_{3})}\]

\[P(E_{3}|A)=\frac{P(E_{3})P(A|E_{3})}{P(E_{1})P(A|E_{1})+P(E_{2})P(A|E_{2}) +P(E_{3})P(A|E_{3})}\]

MEAN AND VARIANCE OF THE DISTRIBUTION

\[Mean(\overline{X})=\sum p_{i}x_{i}\: \: \: where\: \: i=1\: to\: n\]\[Variance(X) = \sum p_{i}x_{i}^{2}\: -\: \left ( \sum p_{i}x_{i} \right )^{2}\: \: \: where\: \: i=1\: to\: n\]


BERNOULLI TRIAL
A trial of a random experiment are called Bernoulli trial, if they satisfy the following conditions:-

a) They are finite in number.

b) They are independent of each other.

c) Each trial has exactly two outcome: success or failure.

d) The probability of success or failure remain same in each trial.

BINOMIAL DISTRIBUTION 
A random variable X which takes values 0,1,2,........,n is said to follow binomial distribution
 if its probability distribution function is given by

\[P(X=r)=\:^{n}C_{r}\: p^{r}q^{n-r}, where r=1,2,3....n\]

Where p and q are the probability of the two events such that   p + q = 1

n is the number of trials and r is the random variable .

The two constants n and p are called parameters of the distribution
 \[P(X=0)+P(X=1)+.....+P(X=n)=^{n}C_{0}p^{0}q^{n-0}+^{n}C_{1}p^{1}q^{n-1}.....^{n}C_{n}p^{n}q^{0}\]
\[\: \: \: \: \: \: =(p+q)^{n}=1^{n}=1\]

Bernoulli’s Trial

Probability of r successes in ‘n’ Bernoulli trial is given by

\[P(r\: successes)=^{n}C_{r}p^{r}q^{n-r}\]

\[P(r\:successes)=\frac{n!}{r!(n-r)!}.p^{r}q^{n-r}\]

Where  n = number of trials

r = number of successful trials = 0, 1, 2, 3, ………., n

p = probability of success in a trial

q = probability of failure in a trial

and  p + q = 1

The probability distribution of number of successes for a random variable X can be written as:

This probability distribution is called Binomial distribution with parameters n and p.

Shortcut Method of finding Mean, Variance and Standard Deviation

The binomial distribution with n Bernoulli trials and success p is also denoted by B(n, p)

Where ‘n’ number of Bernoulli trial

 p denotes the probability of success 

 q denotes the probability of failure then,

Mean =  np

Variance = npq

Standard Deviation = Square root of  npq


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