1.Algebra Booster

8.24 Algebra Booster
72. A is targeting to B, B and C are targeting to A. Probability
of hitting the target by A, B and C are 2/3, 1/2 and
1/3, respectively. If A is hit, find the probability that
hits the target and C does not. [IIT-JEE, 2003]
73. For a student to qualify, he must pass at least two out
of three examinations. The probability that he will pass
the 1 st exam is p. If he fails in one of examinations, the
probability of his passing in the next examinations is
p/2, otherwise it remains the same. Find the probability
that he will qualify? [IIT-JEE, 2003]
74. If three distinct numbers are chosen randomly from the
first 100 natural numbers, the probability that all the
three of them are divisible by both 2 and 3 is
(a) 4/25 (b) 4/35 (c) 4/33 (d) 4/1155.
[IIT-JEE, 2004]
75. A box contains 12 red and 6 white balls. Balls are drawn
from the bag, one at a time, without replacement. If in
6 draws, there are at least 4 white balls, find the probability
that exactly one white ball is drawn in the next
two draws (binomial co-efficients can be left as such).
[IIT-JEE, 2004]
76. If A and B are two independent events, prove that
P(A » B) ◊ P(A¢ »B¢) £ P(C),
where C is an event defined that exactly one of A and B
occurs. [IIT-JEE, 2004]
77. A fair die is rolled. The probability that the first time 1
occurs at the even number of trials is
(a) 6/11 (b) 1/6 (c) 5/36 (d) 5/11
[IIT-JEE, 2005]
78. Rohan goes to office either by a car, a scooter, a bus
or a train, probability of which being 1/7, 3/ 7, 2/7 and
1/7 respectively and probability that he is reaching office
late if he takes a car, a scooter, a bus or a train is
2/9, 1/9, 1/7 and 1/9, respectively. Find the probability
that he has travelled by car, if he reaches office in time.
[IIT-JEE, 2005]
(a)
(c)
n
1 n 1
(c) (d)
n + 1 n + 1 2
Ê 1ˆ
ÁPu ( i ) =
Ë
˜
¯ , the value of P(w/E) is
Comprehensive Link Passage
Paragraph (81 to 83): There are n urns each containing n +
1 balls such that ith urn contains i white balls and (n + 1 – i)
red balls. Let be the event of selecting ith urn, i = 1, 2, 3, …,
n and w denotes the event of getting a white ball.
[IIT-JEE, 2006]
79. If P(u i
) = i, where i = 1, 2, 3, …, n, then lim Pw ( ) is
equal to
(a) 1 (b) 2/3 (c) 3/4 (d) 1/4
80. If P(u i
) = c, where c is a constant, then P(u i
/w) is equal
to
2
(a) (b)
n + 1
81. If n is even and E denotes the event of choosing evennumbered
urn
n
n + 2
n + 2
(b)
2n
+ 1
2( n + 1)
n
1
(d)
n + 1
n + 1
82. One Indian and four American men and their wives are
to be seated randomly around a circular table. The conditional
probability that the Indian man is seated adjacent
to his wife given that each American man is seated
adjacent to his wife is
(a) 1/2 (b) 1/3 (c) 2/5 (d) 1/5
[IIT-JEE, 2007]
83. Let E c denotes the complement of an event E. Let E, F,
G be pair-wise independent events with P(G) > 0 and
P(E « F « G) = 0.
Then P(E c « F c /F) equals
(a) P(E c ) + P(F c ) (b) P(E c ) – P(F c )
(c) P(E c ) – P(F) (d) P(E) – P(F c ).
[IIT-JEE, 2007]
84. An experiment has 10 equally-likely outcomes. Let A
and B be two non-empty events of the same experiment.
If A consists of 4 outcomes, the number of outcomes
that B must have so that A and B are independent,
is
(a) 2, 4 or 8 (b) 3, 6 or 9
(c) 4 or 8 (d) 5 or 10
[IIT-JEE, 2008]
85. Consider the system of equations
ax + by = 0, cx + dy = 0,
where 0 £ a, b, c, d £ 1.
Statement I: The probability that the system of equations
has a unique solution.
Statement II: The probability that the system of equations
has a solution is 1.
[IIT-JEE, 2008]
86. Comprehensive Link Passage
A fair die is tossed repeated until a six is obtained. Let
X denotes the number of tosses required.
(i) The probability that X = 3 equals
(a) 25/216 (b) 25/36
(c) 5/36 (d) 125/216
(ii) The probability that X ≥ 3 equals
(a) 125/216 (b) 25/36
(c) 5/36 (d) 25/216.
(iii) The conditional probability that X ≥ 6 is given X >
3 equals
(a) 125/216 (b) 25/36
(c) 5/36 (d) 25/216.
[IIT-JEE, 2009]
87. Let w be a complex cube root of unity with w π 1. A fair
die is thrown three times.
If r 1
, r 2
and r 3
, are the numbers obtained on the die, the
r
probability that w
1
+ w
2
+ w
3
= 0 is
r
(a) 1/18 (b) 1/9 (c) 2/9 (d) 1/36
[IIT-JEE, 2010]
r