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Q. 18.  Two events A and B will be independent, if

(A)  $\inline A$ and $\inline B$ are mutually exclusive

(B)  $\inline P(A'B')=\left [ 1-P(A) \right ]\left [ 1-P(B) \right ]$

(C)  $P(A)=P(B)$

(D) $P(A)+P(B)=1$

Two events A and B will be independent, if Or   Option B is correct.

Q .17.     The probability of obtaining an even prime number on each die, when a pair of dice is rolled is

(A)  $0$

(B)  $\frac{1}{3}$

(C)   $\frac{1}{12}$

(D)   $\frac{1}{36}$

when a pair of dice is rolled, total outcomes  Even prime number   The probability of obtaining an even prime number on each die                                                                                           Option D is correct.

Q. 16.       In a hostel, $60^{o}/_{o}$ of the students read Hindi newspaper, $40^{o}/_{o}$  read English newspaper and $20^{o}/_{o}$ read both Hindi and English newspapers. A student is selected at random.

(c) If she reads English newspaper, find the probability that she reads Hindi newspaper.

H :   of the students read Hindi newspaper, E :  read English newspaper       and       read both Hindi and English newspapers.  the probability that she reads Hindi newspaper if she reads English newspaper                                                                                                                                                                                           ...

Q. 16    In a hostel, $60\; ^{o}/_{o}$ of the students read Hindi newspaper, $40\; ^{o}/_{o}$  read English newspaper and $20\; ^{o}/_{o}$  read both Hindi and English newspapers. A student is selected at random.

(b) If she reads Hindi newspaper, find the probability that she reads English newspaper.

H :   of the students read Hindi newspaper, E :  read English newspaper       and       read both Hindi and English newspapers. The probability that she reads English newspape if she reads Hindi newspaper                                                                                                                                                                                             ...

Q.16    In a hostel, $\inline 60\; ^{o}/_{o}$  of the students read Hindi newspaper,$\inline 40\; ^{o}/_{o}$ read English newspaper and $\inline 20\; ^{o}/_{o}$ read both Hindi and English newspapers. A student  is selected at random.

(a) Find the probability that she reads neither Hindi nor English newspapers

H :   of the students read Hindi newspaper, E :  read English newspaper       and       read both Hindi and English newspapers. the probability that she reads neither Hindi nor English newspapers                                                                                                                                                                                                      ...

Q.15   One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent ?

(iii) E : ‘the card drawn is a king or queen’

F : ‘the card drawn is a queen or jack’.

One card is drawn at random from a well shuffled deck of  cards Total king or queen = 8 total queen or jack = 8 E : ‘the card drawn is a king or queen’ F : ‘the card drawn is a queen or jack’.  a card which is queen  = 4 Hence, E and F are not indepentdent  events .

Q.15    One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent ?

(ii) E : ‘the card drawn is black’

F : ‘the card drawn is a king’

One card is drawn at random from a well shuffled deck of  cards Total black card = 26 total king =4 E : ‘the card drawn is black’  F : ‘the card drawn is a king’  a card which is black and king = 2 Hence, E and F are indepentdent events .

Q.15    One card is drawn at random from a well shuffled deck of $52$ cards. In which of the following cases are the events $E$ and $F$ independent ?

(i) E : ‘the card drawn is a spade’

F : ‘the card drawn is an ace’

One card is drawn at random from a well shuffled deck of  cards Total ace = 4 total spades =13 E : ‘the card drawn is a spade  F : ‘the card drawn is an ace’  a card which is spade and ace = 1 Hence, E and F are indepentdent events .

Q.14     Probability of solving specific problem independently by A and B are $\frac{1}{2}$  and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that

(ii) exactly one of them solves the problem

and        ,         ,              probability that exactly one of them solves the problem      probability that exactly one of them solves the problem                                                                                                                                                                                                                                    ...

Q.14     Probability of solving specific problem independently by A and B are $\frac{1}{2}$ and  $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that

(i) the problem is solved

and         Since, problem is solved independently by A and B,                                                 probability that the problem is solved

13.     Two balls are drawn at random with replacement from a box containing $10$ black and $8$ red balls. Find the probability that

(iii) one of them is black and other is red.

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Total balls =18 Black balls = 10 Red balls = 8 Let the first ball is black and the second ball is red. The probability of getting a black ball in the first draw                                                                               The ball is replaced after drawing the first ball. The...

Q.13.     Two balls are drawn at random with replacement from a box containing $\inline 10$ black and $\inline 8$ red balls. Find the probability that

(ii)  first  ball is black and second is red.

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Total balls =18 Black balls = 10 Red balls = 8 The probability of getting a black ball in the first draw                                                                               The ball is replaced after drawing the first ball. The probability of getting a red ball in the second draw           ...

Q.13    Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that

(i) both balls are red.

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Total balls =18 Black balls = 10 Red balls = 8 The probability of getting a red ball in first draw                                                                               The ball is repleced after drawing first ball. The probability of getting a red ball in second draw                         ...

Q.12   A die is tossed thrice. Find the probability of getting an odd number at least once.

A die is tossed thrice. Outcomes  Odd numbers  The probability of getting an odd number at first throw                                                                                          The probability of getting an even number                                                                             Probability of getting even number three times                                         ...

Q.11   Given two independent events $\inline A$  and $\inline B$  such that $\inline P(A)=0.3,P(B)=0.6,$  Find

(iv)  $\inline P(neither\; A\; nor\; B)$

Q.11     Given two independent events A and B such that $P(A)=0.3,P(B)0.6,$  Find

(iii) $P(A\; or \; B)$

Q.11     Given two independent events A and B such that $\inline P(A)=0.3,P(B)=0.6,$  Find

(ii)  $\inline P(A \; and \; not\; B)$

Given two independent events and .

Q.11   Given two independent events $\inline A$ and $\inline B$ such that $\inline P(A)=0.3,P(B)=0.6,$  Find

(i)  $\inline P(A \; and\; B)$

Given two independent events and .         Also , we know

Q.10    Events A and B are such that $P(A)=\frac{1}{2},P(B)=\frac{7}{12}$  and $P(not \; A \; or\; not\; B)=\frac{1}{4}.$  State whether $A$ and $B$ are independent ?

If   and  are two events such that  and                                                                                                                                                                           As we can see  Hence, A and B are not independent.

Q.9 If $A$  and $B$ are two events such that $P(A)=\frac{1}{4},P(B)=\frac{1}{2}$ and $P(A\cap B)=\frac{1}{8},$   find  $P(not\; A\; and\; not\; B).$

If   and  are two events such that  and                                            use,
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