:
In everyday life,we come across statements such as:
It will probably rain today
I doubt that he will pass the test.
Most probably ,kavita will stand first in the annual examination.
Chances are high that the prices of diesel will go high first.
There is a 50 - 50 chances of India winning a toss in todays match.
The words like 'probably' , 'doubt','most probably','chances' used in the statements above involves an element of uncertanity.
For Example:
'probably rain' means may be it may or maynot rain today.
Here we are predicting rain today based on our past experience.
The uncertanity of 'probably' etc can be measured by mean of 'probabilty' in some cases
Definition of Probability
- Probability is defined as chance of occurence of a certain event when it is expressed quantitavitely: It is a number which has the value from 0 to 1.
- The probability of an event which cannot occur is 0 whereas an event which is certain to occur has probability 1.
In probability we use words like coin , die , cards (plating cards ) etc
coin : It is a piece of stamped metal used as money.It has two faces known as Head (H) and Tails (T) .
If we toss a coin ,we shall get Head or Tail.
Dice : It is a small cube used for playing some indoor games like ludo etc.It is also used in gambling .It has six plane faces mark dot as 1 to 6.
Cards :A pack of cards has 52 cards. These 52 cards are divided into four suits , having 13 cards in each suits.
These suits with color of faces are under as:
suit color of faces
spades black faced
clubs black faced
Hearts red faced
Diamonds red faced
9 of the 12 cards are numbered from 2 to 10 and the remaining four cards are as:
an ace, a king, a queen, a jack
(i) Face cards : The aces ,the kings , the queen ,and the jack are called face cards.
(ii)Court cards: Court cards consist of the kings ,the queen , and the jacks are these are 12 in number.
Probability: Dice Problem
Example :six dice are thrown simultaneously .Find the probability that
(i) all of them show the same face.
(ii)all of them show different faces.
(iii)exactly three of them show the same face and remaining three show different faces
(iv)atleast four of them show the same face.
Solution 1 :
(i) The total number of elementary events associated to the random experiment of throwing sixdice is 6x 6 x 6 x 6 x 6 x 6 = 66
(ii) All dices show the same face mean we are getting same number on all six dice.The number of ways for which is 6C1
Hence ,Required probability = 6C1/66
(iii)Select a number which occur on three dice out of six numbers 1,2,3,4,5, 6.This can be done 6C1
ways.Now select three number out of remaining 5 numbers.This can be done in 5C3 ways.Now we have
6 numbers like 1,2,3,4,4,4,;2,3,6,1,1,1k etc .These digit can be arranged in 6!/3!.
So the number of ways in which three dice show the same face and remaining three shows has distinct
faces is
6C1 x 5C3 x 6!/3!
Hence ,required probability = 6C1 x 5 C3 x 6! / 3! / 6 6
(iv)Atleast four of them show the same face means either four dice show same face and remaining two
show distinct faces or five dice show the sme face and remaining one shows a different faces or all six
show the same faces .
so required probabillity= 6C1 x 5 C2 x 6!/4! +6C1 x 5C1 x 6!/5! + 6C1 / 6 6
Probability : Random Experiment
Example: The digit 1 , 2 , 3 , 4 , 5, 6, 7 , 8, and 9 are written in random order form a nine digit
number.Find the probability that this number isdivisible by 4
Solution :
There are 9 p 9 = 9! ways of arranging the given digit to form a nine digit number.So,the
total number of 9 digit formed by the given digit = 9 !
Out of these 9! numbers only those number are divisible by 4 which have their last 4 digit as even
natural number and the number formed by their last two digit are divisible by 4.
The various possiblities of last two digit are:
12 , 32 , 52 , 72 , 92 , 24 , 64 , 84
16 , 36 , 56 , 76 , 96 , 28 , 48 , 68
This means that there are 16 ways of choosing the last two digit .
Corresponding to each of these ways the remaining 7 digit can be arranged in 7 p 7 = 7 ! ways.
Therefore , the total number of 9 digit numbers divisible by 4 is 16 x 7 !
Hence ,required probability = 16 x 7 !/ 9 != 2/9
Related Tags
Introduction to probability, Understanding about probability, Study of probability