Thursday, 19 August 2021

Probability

1.1 EXPERIMENT : The word experiment means an operation, which can produce well defined outcomes. 
The are two types of experiment : 
                   (i) Deterministic experiment 
                    (ii) Probabilistic or Random experiment 

(i) Deterministic Experiment : Those experiment which when repeated under identical conditions, produced the same results or outcome are known as deterministic experiment. For example, Physics or Chemistry experiments performed under identical conditions.

 (ii) Probabilistic or Random Experiment :- In an experiment, when repeated under identical conditions donot produce the same outcomes every time. For example, in tossing a coin, one is not sure that if a head or tail will be obtained. So it is a random experiment. 

Sample space : The set of all possible out comes of a random experiment is called a sample space associated with it and is generally denoted by S. 
For example, When a dice is tossed then S = {1, 2, 3, 4, 5, 6}.

 Event : A subset of sample space associated with a random experiment is called an event. For example, In tossing a dive getting an even no is an event. 

Favorable Event : Let S be a sample space associated with a random experiment and A be event associated with the random experiment. The elementary events belonging to A are know as favorable events to the event A. 
For example, in throwing a pair of dive, A is defined by “Getting 8 as the sum”. Then following elementary events are as out comes : (2, 6), (3, 5), (4, 4) (5, 3) (6, 2). So, there are 5 elementary events favorable to event A. 

1.2 PROBABILITY : If there are n elementary events associated with a random experiment and m of them are favorable to an event A, then the probability of happening or occurrence of event A is denoted by P(A)
 Thus,
    

P(A)     =   (Total number of favourable outcomes)/(Total number of possible outcomes)

                   =  m/n 

And 0 £ P(A) £ 1

 If, P(A) = 0, then A is called impossible event

If, P(A) = 1, then A is called sure event

                                P(A) + P (A") = 1

Where P(A) = probability of occurrence of A. 

            P (A') = probability of non - occurrence of A.

 

Ex.1 A box contains 5 red balls, 4 green balls and 7 white balls. A ball is drawn at random from the box. Find the probability that the ball drawn is (i) white (ii) neither red nor white

Sol. Total number of balls in the bag = 5 + 4 + 7 = 16

\ Total number of elementary events =16

(i)                There are 7 white balls in the bag.

\ Favorable number of elementary events = 7

P(Getting a white ball ) = (Total No. of elementaryevents) /(Total No. favourable elementaryevents)

                    = 7 /16

 (ii) There are 4 balls that are neither red nor white 


 Favorable number of elementary events = 4 

Hence, P (Getting neither red not white ball) = 4/16 =1/4

 Ex.2 All the three face cards of spades are removed from a well-shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting  [CBSE - 2007] (i) black face card                     (ii) a queen             (iii) a black card.

 Sol. After removing three face cards of spades (king, queen, jack) from a deck of 52 playing cards, there are 49 cards left in the pack. 

Out of these 49 cards one card can be chosen in 49 ways. 

Therefore, Total number of elementary events = 49 

(i) There are 6 black face cards out of which 3 face cards of spades are already removed. 

So, out of remaining 3 black face cards one black face card ban be chosen in 3 ways. 

Therefore, Favorable number of elementary events = 3 

Hence, P (Getting a black face card ) = 3/49

 (ii) There are 3 queens in the remaining 49 cards.

 So, out of these three queens, on queen can be chosen in 3 ways 

Therefore, Favorable number of elementary events = 3

 Hence P (Getting a queen) = 3/49

 (iii) There are 23 black cards in the remaining 49 cards, 

So, out to these 23 black card, one black card can be chosen in 23 ways 

Therefore, Favorable number of elementary events = 23 

Hence, P (Getting a black card) =  23/49

 Ex.3 A die is thrown, Find the probability of (i) prime number (ii) multiple of 2 or 3 (iii) a number greater than 3.

Sol.

In a single throw of die any one of six numbers 1,2,3,4,5,6 can be obtained. 

Therefore, the tome number of elementary events associated with the random experiment of throwing a die is 6.

 (i) Let A denote the event “Getting a prime no”. 

Clearly, event A occurs if any one of 2,3,5 comes as out come. 

Therefore, Favorable number of elementary events = 3 

Hence, P (Getting a prime no.) = 3/6=1/2

(ii) An multiple of 2 or 3 is obtained if we obtain one of the numbers 2,3,4,6 as out comes.

Therefore, Favorable number of elementary events = 4 

Hence, P (Getting multiple of 2 or 3) = 4/6=2/3

 (iii) The event “Getting a number greater than 3” will occur, if we obtain one of number 4,5,6 as an out come. 

Therefore, Favorable number of out comes = 3

 Hence, required probability = 3/6=1/2

 Ex.4 Two unbiased coins are tossed simultaneously. Find the probability of getting (i) two heads (ii) at least one head (iii) at most one head. 

Sol

If two unbiased coins are tossed simultaneously, 

we obtain any one of the following as an out come : HH, HT, TH, TT 

Therefore, Total number of elementary events = 4

 (i) Two heads are obtained if elementary event HH occurs. 

Therefore, Favorable number of events = 1 Hence, P (Two heads) =  1/4

 (ii) At least one head is obtained if any one of the following elementary events happen : HH, HT, TH 

Therefore, favorable number of events = 3 

Hence P (At least one head) = 3/4

 (iii) If one of the elementary events HT, TH, TT occurs, than at most one head is obtained 

Therefore, favorable number of events = 3 

Hence, P (At most one head) = 3/3

Ex.5 A box contains 20 balls bearing numbers, 1,2,3,4…...20. A ball is drawn at random from the box. What is the probability that the number of the ball is (i) an odd number (ii) divisible by 2 or 3 (iii) prime number 

Sol. Here, total numbers are 20. 

 Total number of elementary events = 20 

(i) The number selected will be odd number, if it is elected from 1,3,5,7,9,11,13,15,17,19 

 Therefore, Favorable number of elementary events = 10 

Hence, P (An odd number ) = 10/20=1/2

 (ii) Number divisible by 2 or 3 are 2,3,4,6,8,9,10,12,14,15,16,18,20 

Therefore, Favorable number of elementary events = 13

 P (Number divisible by 2 or 3) =  13/20

 (iii) There are 8 prime number from 1 to 20 i.e., 2,3,5,7,11,13,17,19

Therefore, Favorable number of elementary events = 8 

P (prime number ) = 8/20=2/5

 Ex.6 A die is drop at random on the rectangular region as shown in figure. What is the probability that it will land inside the circle with diameter 1m ? 

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