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-1

PROBABILITY 
For classes 8th, 9th, 10th, 11th

Basic concepts of probability useful for classes 9, 10 and 11. Explanation with playing cards, tossing one coin, two coins and three coins.


Probability of an event 

      =  


If E is an event then :-  P(E) + P(notE) = 1
Probability of an event is greater than, equal to 0 but is less than, equal to 1. This is also be called as the range of Probability

\[0\leq P(E)\leq 1\]

Impossible Event:- 
An event which never occur is called impossible event. 
P(Impossible Event) = 0

Sure Event :
An even which happens for sure is called sure event. Sure event is also be called as the certain event.
P(Sure Event or Certain Event) = 1

Equally likely events:- 
If every event of the sample space has equal possibility to happen, then the events are called equally likely events. (e.g.  Tossing a coin)

PROBABILITY WITH PLAYING CARDS

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DESCRIPTION OF PLAYING CARDS

Total playing cards

52

Number of black cards

26

Number of red cards

26

Number of spade card (♠) 

13

Number of Clubs card(♣) 

13

Number of diamond card()

13

Number of heart card(

13

Number of king card

4

Number of queen card

4

Number of Jack card

4

Number of ace card

4

Number of face card (Jack, Queen, King)

12


PROBABILITY WITH  TOSSING OF COINS

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When one coin is tossed then sample space
S = {H, T} ⇒ n(S) = 2

When two coins are tossed then sample space
S = {HH, HT, TH, TT}  ⇒ n(S) = 4 = {1, 2, 1}

When three coins are tossed then sample space
S = {HHH,HHT, HTH, THH, HTT, THT, TTH, TTT}
 ⇒ n(S) = 8 = {1, 3, 3, 1}

When three coins are tossed then sample space
S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, THHT, THTH, TTHH, HTTH, HTTT, THTT, TTHT, TTTH, TTTT}
 ⇒ n(S) = 16 = {1, 4, 6, 4, 1} 

Formula for finding the sample space when number of coins are tossed = 

Where n is the number of trials


PROBABILITY WITH TOSSING OF DICE
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When a die is thrown then sample space
S = {1,2,3,4,5,6} ⇒ n(S) = 6

When two dice are thrown then sample space
S = {(1,1), (1,2), 1,3), 1,4), 1,5), 1,6), 
       (2,1), (2,2), 2,3), 2,4), 2,5), 2,6), 
      (3,1), 3,2), (3,3), 3,4), 3,5), 3,6),
      (4,1), (4,2), 4,3), 4,4), 4,5), 4,6), 
       (5,1), 5,2), 5,3), 5, 4), 5, 5), 5, 6),
        (6,1), (6,2), 6,3), (6,4), 6,5), 6,6)} 
⇒ n(S) = 36
Number of doublets in this sample space = 6 

{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} are called doublets

When three dice are  thrown then.
Number of elements in sample space = n(S) = 216

Formula for finding the sample space when number of coins are tossed = 

Where n is the number of trials


Where  n is the number of trials

Al least ⇒  ≥ (greater than or equal to)

At most ⇒  ≤ (less than or equal to)

If there is "or" between two or more events then the probability of different events are added.

If there is "and" between two or more events, then first we find the common events and then required probability is the probability of common events.
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Important Explanation

Question: When two dice are thrown then what is the probability such that sum of numbers on two dice is  2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
Solution:  This question can be explained from the following table
Sum of the numbers on two dice
Observations
No. of observations
Probability
2
(1,1)
1
1/36
3
(1,2), (2,1)
2
2/36
4
(1,3), (2,2), (3,1)
3
3/36
5
(1,4), (2, 3), (3,2), (4,1)
4
4/36
6
(1,5), (2,4), (3,3), (4,2), (5,1)
5
5/36
7
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
6
6/36
8
(2.6), (3,5), (4,4), (5,3), (6,2)
5
5/36
9
(3, 6),(4, 5), (5,4), (6, 3)
4
4/36
10
(4,6), (5,5), (6,4)
3
3/36
11
(5,6), (6,5)
2
2/36
12
(6,6)
1
1/36
Two solve this problem first we write all 36 observations  obtained after throwing two dice, then find the number of observations having sum 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, then find their probability.
 This can be made easy if we follow the figure below. we understand it as 

No of observations with sum 2 = 1
P( Getting sum 2) = 1/36

No of observations with sum 3 = 2
P( Getting sum 3) = 2/36

No of observations with sum 7 = 6
P( Getting sum 7) = 6/36

No of observations with sum 11 = 2
P( Getting sum 11) = 2/36

........................and so on
Probability
Next topics are for 10+1 and 10+2 class students 

Simple event

If an event have only one sample point of sample space, it is called a simple event.
For example : HH, HHH, TT, TTT, 222, all are called simple events.

Compound Event:

If an event have more than one sample point in a sample space then it is called a compound event.
For Example:  HT, THH, 234, etc all are compound event.

Mutually Exclusive events:

If A and B are two events and A⋂B = Φ, then A and B are called mutually exclusive events.

Exhaustive events:

If union of all the events is equal to the sample space then the events are called exhaustive events.

Consistency of Probability:

If A and B are two events then probability is said to be consistently defined if 
P(AUB) > P(A) and P(AUB) > P(B)

Other Important Results on Probability

P(A or B) = P(A U B) = P(A) + P(B) - P(A⋂ B)

P(Either A or B) = P(AUB)

P(A and B) = P(A ⋂ B)

P(Neither A nor B) = P(A' ⋂ B') = P(AUB)' = 1 - P(AUB)

P(not A) = P(A') = 1- P(A)

P(not A or not B) = P(A' U B') = P(A ⋂ B)' = 1 - P(A ⋂ B)

P(A⋂B') = P(A - B) = P(A) - P(A⋂B)

P(A'⋂B) = P(B - A) = P(B) - P(A⋂B)

If A and B are two events then  P(At least one of them ) = P(AUB)

INTEGRATION OF PROBABILITY WITH ARTIFICIAL INTELLIGENCE TOOL


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