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The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
(A) 0 (B) 
(C) 
(D) 
When two dice are rolled, the number of outcomes is 36.
The only even prime number is 2.
Let E be the event of getting an even prime number on each die.
∴ E = {(2, 2)}
Therefore, the correct answer is D.
If
, find P (A ∩ B) if A and B are independent events.
If A &B are independent event P(A^B)=PA×P,B=1/3×1/5=1/15
Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.
Probability of getting a black card =26/52=1/2 and probability of getting black cards =1/2×25/51=25/102
A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.
Box contains 12 good and 3 bad oranges if 3 oranges selected are good once the box is approved for sale. Probability of selected 3 good oranges =12×11×10/15×14×13=44/91.
A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not.
If a fair coin and an unbiased die are tossed, then the sample space S is given by,
Let A: Head appears on the coin
B: 3 on die 
∴
Therefore, A and B are independent events.
A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, ‘the number is even,’ and B be the event, ‘the number is red’. Are A and B independent?
Probability of getting even number P(A)=3/6=1/2 and probability of getting red colour P(B)=1/2 P(A^B)=1/6 is not equal to P(A)×,P(B)=1/2×1/2=1/4 therefore A and B are not independent
Let E and F be events with
. Are E and F independent?
P(E) × p(F)=3/10×3/5=9/50 P(E intersection F) is not equal toP(E)×P(F) there for event E and F are not independently
Given that the events A and B are such that P(A) = , P(A ∪ B) =
and P(B) = p. Find p if they are (i) mutually exclusive (ii) independent.
It is given that
(i) When A and B are mutually exclusive, A ∩ B = Φ
∴ P (A ∩ B) = 0
It is known that, 
(ii) When A and B are independent, 
It is known that,
Let A and B be independent events with P (A) = 0.3 and P (B) = 0.4. Find
(i) P (A ∩ B) (ii) P (A ∪ B)
(iii) P (A|B) (iv) P (B|A)
It is given that P (A) = 0.3 and P (B) = 0.4
(i) If A and B are independent events, then
(ii) It is known that, 
(iii) It is known that, 
(iv) It is known that, 
If A and B are two events such that
, find P (not A and not B).
P(A)=1/4 P(B)=1/2 P(A intersect B)=1/8
P(A U B)=P(A)+P(B)-P(A intersect B)
=(1/4)+(1/2)-(1/8)=(2+4-1)/8=5/8
P(not A intersect not B)= 1- P(A U B)
= 1- (5/8)= 3/8
P(not A)= 1-P(A)=3/4
P(not B)=1-P(B)=1/2
P(not A U not B)= p(not A)+P(not B)-P(not A intetrsect not B)
=(3/4)+(1/2)-(3/8)=(6+4-3)/8=7/8
Ans 7/8.
Events A and B are such that
. State whether A and B are independent?
It is given that 
Therefore, A and B are not independent events.
Given two independent events A and B such that P (A) = 0.3, P (B) = 0.6. Find
(i) P (A and B) (ii) P (A and not B)
(iii) P (A or B) (iv) P (neither A nor B)
i) P(A and B)= 0.3 × 0.6=0.18
ii)P(A and not B) = 0.3×0.4=0.12
iii) P(A or B)=0.3+0.6-0.18=0.72
iv) P(neither A nor B) =1- P(A or B) =1-0.72=0.28
A die is tossed thrice. Find the probability of getting an odd number at least once.
Probability of getting an odd number in a single throw of a die = 
Similarly, probability of getting an even number =
Probability of getting an even number three times =
Therefore, probability of getting an odd number at least once
= 1 − Probability of getting an odd number in none of the throws
= 1 − Probability of getting an even number thrice
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.
(ii) first ball is black and second is red.
(iii) one of them is black and other is red.
Total number of balls = 18
Number of red balls = 8
Number of black balls = 10
(i) Probability of getting a red ball in the first draw =
The ball is replaced after the first draw.
∴ Probability of getting a red ball in the second draw =
Therefore, probability of getting both the balls red =
(ii) Probability of getting first ball black =
The ball is replaced after the first draw.
Probability of getting second ball as red =
Therefore, probability of getting first ball as black and second ball as red =
(iii) Probability of getting first ball as red =
The ball is replaced after the first draw.
Probability of getting second ball as black =
Therefore, probability of getting first ball as black and second ball as red = 
Therefore, probability that one of them is black and other is red
= Probability of getting first ball black and second as red + Probability of getting first ball red and second ball black
Probability of solving specific problem independently by A and B are
respectively. If both try to solve the problem independently, find the probability that
(i) the problem is solved (ii) exactly one of them solves the problem.
Probability of solving the problem by A, P (A) =
Probability of solving the problem by B, P (B) =
Since the problem is solved independently by A and B,
Probability that the problem is solved = P (A ∪ B)
= P (A) + P (B) − P (AB)
(ii) Probability that exactly one of them solves the problem is given by, 
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’
(ii) E: ‘the card drawn is black’
F: ‘the card drawn is a king’
(iii) E: ‘the card drawn is a king or queen’
F: ‘the card drawn is a queen or jack’
Total cards=52 probability of getting space card p(E)=13/52=1/4 there are 4 aces in a deck of cards probability of getting a ace p(E)=4/52=1/13 there is only one card which is an ace ofspade =1/52 =p(E^F) =1/4×1/13=1/52=P(A)×P(B) event E and F are independent
In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English news papers. A student is selected at random.
(a) Find the probability that she reads neither Hindi nor English news papers.
(b) If she reads Hindi news paper, find the probability that she reads English news paper.
(c) If she reads English news paper, find the probability that she reads Hindi news paper.
Let H denote the students who read Hindi newspaper and E denote the students who read English newspaper.
It is given that,
Probability that a student reads Hindi or English newspaper is, 
(ii) Probability that a randomly chosen student reads English newspaper, if she reads Hindi news paper, is given by P (E|H).
(iii) Probability that a randomly chosen student reads Hindi newspaper, if she reads English newspaper, is given by P (H|E).
Two events A and B will be independent, if
(A) A and B are mutually exclusive
(B) 
(C) P(A) = P(B)
(D) P(A) + P(B) = 1
Two events A and B are said to be independent, if P(AB) = P(A) × P(B)
Consider the result given in alternative B.
This implies that A and B are independent, if
Distracter Rationale
A. Let P (A) = m, P (B) = n, 0 < m, n < 1
A and B are mutually exclusive.
C. Let A: Event of getting an odd number on throw of a die = {1, 3, 5}
B: Event of getting an even number on throw of a die = {2, 4, 6}
Here,
D. From the above example, it can be seen that,
However, it cannot be inferred that A and B are independent.
Thus, the correct answer is B.
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