JEE Mathematics - Probability Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
Out of (2n+1) tickets consecutively numbered, three are drawn at random. The probability that numbers are in A.P., is
- (a)
\(\cfrac { 1 }{ 4{ n }^{ 2 }-1 } \)
- (b)
\(\cfrac { 2 }{ 4{ n }^{ 2 }-1 } \)
- (c)
\(\cfrac { 3n }{ 4{ n }^{ 2 }-1 } \)
- (d)
\(\cfrac { 4n }{ 4{ n }^{ 2 }-1 } \)
Ten persons are sitting on a round table. The probability of three particular persons, sitting together, is
- (a)
\(\cfrac { 5 }{ 12 } \)
- (b)
\(\cfrac { 7 }{ 12 } \)
- (c)
\(\cfrac { 11 }{ 12 } \)
- (d)
\(\cfrac { 1 }{ 12 } \)
A, B, C are three events, such that P(B)=\(\cfrac { 3 }{ 4 } \) ,\(P(A\cap B\cap \overline { C } )=\cfrac { 1 }{ 3 } \) and \(P(\overline { A } \cap B\cap \overline { C } )=\cfrac { 1 }{ 3 } \) then \(P(B\cap C)\) equals
- (a)
\(\cfrac { 1 }{ 12 } \)
- (b)
\(\cfrac { 1 }{ 6 } \)
- (c)
\(\cfrac { 1 }{ 15 } \)
- (d)
\(\cfrac { 1 }{ 9 } \)
Two dice are tossed. The probability that the total score is a prime number is
- (a)
\(\cfrac { 1 }{ 6 } \)
- (b)
\(\cfrac { 5 }{ 12 } \)
- (c)
\(\cfrac { 1 }{ 2 } \)
- (d)
NONE OF THESE
An anti-aircraft gun takes a maximum of four shots at an enemy plane moving away from it. The probability of hitting the plane at the first, second, third and forth shot are 0.4, 0.3, 0.2, 0.1 respectively. The probability that the gun hits the plane is
- (a)
0.24
- (b)
0.21
- (c)
.16
- (d)
0.6976
A and B take turns in tossing a pair of dice. The first to get a throw of 7 wins. If A starts the game the chance of winning of A, is
- (a)
\(\cfrac { 6 }{11 } \)
- (b)
\(\cfrac { 5 }{11 } \)
- (c)
\(\cfrac { 1 }{11 } \)
- (d)
NONE OF THESE
If 10% of bolts produced by a machine are defective, then the probability that out of a sample selected at random, of 7 bolts, not more than 1 is defective is
- (a)
\({ (1.6) }^{ 6 }{ (0.9) }^{ 4 }\)
- (b)
\({ (0.9) }^{ 6 }{ (1.6) }^{ }\)
- (c)
\({ (1.6) }^{ 4}{ (0.9) }^{ 6 }\)
- (d)
NONE OF THESE
The least number of times a fair coin must be tossed so that probability of getting at least head is not less than 0.8 is,
- (a)
3
- (b)
4
- (c)
5
- (d)
6
A coin is tossed 7 times. Each time a man calls head. The probability that he wins the toss on at least 4 occasions is
- (a)
1/4
- (b)
5/8
- (c)
1/2
- (d)
None of these
Observe the following columns
Column I | Column II |
---|---|
A.The probability that A,B,C solve a problem independently is \({1\over2},{1\over 3}\) and \(1\over4\). If the probability that the problem will be solved is \(\lambda\) and that the problem is solved by only one of them is \(\mu\) . then |
p. \(\lambda+\mu={13\over24}\) |
B. The probability of hitting a target by three marks men is \({1\over2},{1\over 3}\) and \(1\over4\). respectively. If the probabilty that exatly two of them will hit the target is \(\lambda\) and that at least two of them hit the target is\(\mu\) , then |
q. \(\lambda+\mu={29\over24}\) |
C. A bag contains 4 white and 2 black balls. Another contains 3 white and 5 black balls. One ball is drawn from each bag. If the probability that both are black is \(\lambda\) and that both are white is , \(\mu\) then |
r. \(\lambda+\mu={11\over24}\) |
S.\(\lambda-\mu={7\over24}\) | |
t. \(\mu-\lambda={1\over24}\) |
- (a)
qs, pt, rt
- (b)
sp, st, qt
- (c)
tq, ts, tr
- (d)
None of these
A man draws a card from a pack of 52 playing cards, replaces it and shuffles the pack. He continues this processes until he gets a card of spade. The probability that he will ail the first two times is
- (a)
\(9\over16\)
- (b)
\(1\over16\)
- (c)
\(9\over 64\)
- (d)
None of these
A twelve sided die A has 9 white faces and 3 black faces, whereas another twelve sided die B has 3 white and 9 black aces. A fair coin is tossed once. If it falls head, a series of throws is made with die A alone, if it falls tail, then only die B is used. Then, the probability that E1 : even denoting that coin falls head; E2: event denoting that coin alls tail; E: event denoting that first throw of dice yields a white face
- (a)
white face turns up at the first throw is 1/4
- (b)
first two throws yield white face is 11/16
- (c)
first two throws yield white face is 5/16
- (d)
None of the above
It is given that the events A and B such that \(P(A)={1\over4}\), P(A/B)=\(1\over2\) and P(B/A)=\({2\over3}\). Then, P(B) is
- (a)
\(1\over2\)
- (b)
\(1\over6\)
- (c)
\(1\over3\)
- (d)
\(2\over3\)
Two aeroplanes I and II target a bomb in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only, if the first misses the target is hit by the second plane, is
- (a)
0.06
- (b)
0.14
- (c)
0.32
- (d)
0.7
The altitude through A of \(\Delta \) ABC meets BC at D and the circumscribed circle at E. If D \(\equiv \) (2,3), E \(\equiv \) (5, 5), the ordinate of the orthocentre being a natural number, the probability that the orthocentre lies on the lines
y = 1
y = 2
y = 3
.............
.............
.............
y = 10 is
- (a)
\(\frac { 2 }{ 5 } \)
- (b)
\(\frac { 3 }{ 5 } \)
- (c)
\(\frac { 4 }{ 5 } \)
- (d)
\(\frac { 1 }{ 7 } \)
In a game called "odd man out", n (n > 2) persons toss a coin to determine who will buy refreshments for the entire group. A person who gets an outcome different from that of the rest of the members of the group is called the odd man out. If the probability that there is a loser in any game is \(\frac { 1 }{ 2 } \) , then the value of n is
- (a)
4
- (b)
7
- (c)
8
- (d)
11
For n independent events A/5, P(Ai) = 1 / (1 + i), i = 1, 2, ... , n. The probability that atleast one of the events occurs is
- (a)
1/n
- (b)
1/(n+1)
- (c)
n/(n+1)
- (d)
none of these
The probabilities that a student will obtain grades A, B, C or D are 0.30, 0.35, 0.20 and 0.15 respectively. The probability that he will receive atleast C grade, is
- (a)
0.65
- (b)
0.85
- (c)
0.80
- (d)
0.20
A man is known to speak truth in 75% cases. If he throws an unbiased die and tells his friend that it is a six, then the probability that it is actually a six, is
- (a)
1/6
- (b)
1/8
- (c)
3/4
- (d)
3/8
A bag contains 5 red, 3 white and 2 black balls. If a ball is picked at random, the probability that it is red, is
- (a)
1/5
- (b)
1/2
- (c)
3/10
- (d)
9/10
Three players A, B, C in this order, cut a pack of cards, and the whole pack is reshuffled after each cut. If the winner is one who first draws a diamond, then C's chance of winning is
- (a)
9/28
- (b)
9/37
- (c)
9/64
- (d)
27/64
An unbiased die with faces marked 1, 2, 3, 4, 5 and 6 is rolled four times. Out of the four face values obtained, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5, is
- (a)
1/81
- (b)
16/81
- (c)
65/81
- (d)
80/81
One ticket is selected at random from 100 tickets numbered 00, 01, 02, ... ,99. Suppose X and Y are the sum and product of the digit found on the ticket P (X = 7 / Y = 0) is given by
- (a)
2/3
- (b)
2/19
- (c)
1/50
- (d)
none of these
10% bulbs manufactured by a company are defective. The probability that out of a sample of 5 bulbs, none is defective, is
- (a)
\(\left( 1/2 \right) ^{ 5 }\)
- (b)
\(\left( 1/10 \right) ^{ 5 }\)
- (c)
\(\left( 9/10 \right) \)
- (d)
\(\left( 9/10 \right) ^{ 5 }\)
Suppose X is a binomial variate B (5, p) and P(X = 2) = P(X = 3), then p is equal to
- (a)
1/2
- (b)
1/3
- (c)
1/4
- (d)
1/5
\({ x }_{ 1 },{ x }_{ 2 },{ x }_{ 3 },...,{ x }_{ 50 }\) are fifty real numbers such that \({ x }_{ r }<{ x }_{ r+1 }\) for r = 1,2, 3, ... ,49. Five numbers out of these are picked up at random. The probability that the five numbers have x20 as the middle number is
- (a)
\(\frac { ^{ 20 }{ { C }_{ 2 }\times ^{ 30 }{ { C }_{ 2 } } } }{ ^{ 50 }{ { C }_{ 5 } } } \)
- (b)
\(\frac { ^{ 30 }{ { C }_{ 2 }\times ^{ 19 }{ { C }_{ 2 } } } }{ ^{ 50 }{ { C }_{ 5 } } } \)
- (c)
\(\frac { ^{ 19 }{ { C }_{ 2 }\times ^{ 30 }{ { C }_{ 3 } } } }{ ^{ 50 }{ { C }_{ 5 } } } \)
- (d)
none of these
If a \(\in \) [-20,0], then the probability that the graph of the function y = 16x2+8(a+5)x-7a-5 is strictly above the x-axis is
- (a)
1/2
- (b)
1/17
- (c)
17/20
- (d)
none of these
A second order determinant is written down at random using the numbers 1, - 1 as elements. The probability that the value of the determinant is non zero is
- (a)
1/2
- (b)
3/8
- (c)
5/8
- (d)
1/3
A five digit number is chosen at random. The probability that all the digits are distinct and digits at odd place are odd and digits at even places are even is
- (a)
1/25
- (b)
25/567
- (c)
1/37
- (d)
1/74
A die is thrown 2n + 1 times, n \(\in \) N. The probability that faces with even numbers show odd number of times is
- (a)
\(\frac { 2n+1 }{ 2n+3 } \)
- (b)
less than \(\frac { 1 }{ 2 } \)
- (c)
greater than 1/2
- (d)
none of these
Three six faced fair dice are thrown together. The probability that the sum of the numbers appearing on the dice is k(3\(\le \)k\(\le \)8) is
- (a)
\(\frac { (k-1)(k-2) }{ 432 } \)
- (b)
\(\frac { k(k-2) }{ 432 } \)
- (c)
\(^{ k-1 }{ { C }_{ 2 } }\times \frac { 1 }{ 216 } \)
- (d)
\(\frac { k^{ 2 } }{ 432 } \)
Three numbers are chosen at random without replacement from {1, 2, 3, ... ,10}. The probability that minimum of the chosen number is 3 or their maximum is 7, cannot exceed
- (a)
\(\frac { 11 }{ 30 } \)
- (b)
\(\frac { 11 }{ 40 } \)
- (c)
\(\frac { 11 }{ 50 } \)
- (d)
\(\frac { 11 }{ 60 } \)
A random variable X takes values 0, 1, 2, 3, ... with probability proportions to \((x+1)\left( \frac { 1 }{ 5 } \right) ^{ x },\) then
- (a)
\(P(X=0)=\frac { 16 }{ 25 } \)
- (b)
\(P(X\le 1)=\frac { 112 }{ 125 } \)
- (c)
\(P(X\le 1)=\frac { 112 }{ 125 } \)
- (d)
\(E(X)=\frac { 25 }{ 32 } \)
If n positive integers taken at random and multiplied together, then the chance that the last digit of the product would be
2,4,5 or 8 is
- (a)
\(\left( \frac { 4 }{ 10 } \right) ^{ n }\)
- (b)
\(\frac { { 5 }^{ n }-{ 4 }^{ n } }{ { 10 }^{ n } } \)
- (c)
\(\frac { { 4 }^{ n }-{ 2 }^{ n } }{ { 5 }^{ n } } \)
- (d)
\(\left( \frac { 1 }{ 2 } \right) ^{ n }\)
The set A-B denotes the event
- (a)
A and B
- (b)
A or B
- (c)
only A
- (d)
A but not B
Consider the experiment of rolling a die. Let A be the event of 'getting a prime number and B be the event of 'getting an odd number', then
A or B=
- (a)
{1, 2, 3, 4}
- (b)
{1, 2, 3, 5}
- (c)
{3, 5}
- (d)
{1, 3, 5}
Consider the experiment of rolling a die. Let A be the event of 'getting a prime number and B be the event of 'getting an odd number', then
A and B=
- (a)
{1, 2, 3, 5}
- (b)
{1, 2}
- (c)
{3, 5}
- (d)
{5}
Two dice are thrown. The events P, Q and R are described as follows:
P: getting an odd number on the first die.
Q: getting an even number on the first die.
R: getting atmost 6 as sum of the numbers on two dice.
The set of the event (Q and R) is
- (a)
{(2, 1), (2,2), (2, 3)}
- (b)
{(3, 1),(4, 1)}
- (c)
{(2, 1), (2,2)(2,3), (4, 1)}
- (d)
None of these
Two dice are thrown and the sum of the numbers which come up on the dice is noted. Let us consider the following events associated with this experiment.
A: "the sum is even".
B: "the sum is a multiple of3".
C: "the sum is less than 4".
D: "the sum is greater than 11".
Which pair of these events is mutually exclusive?
- (a)
A and B
- (b)
B and C
- (c)
C and D
- (d)
A and C
Two dice are thrown. The events A, Band C are as
follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice \(\le \)5.
Then, which of the following is true?
(i) A and B are mutually exclusive.
(ii) A and B are mutually exclusive and exhaustive.
(iii) A = B'
(iv) A and C are mutually exclusive.
(v) A and B' are mutually exclusive.
(vi) A', B' and C are mutually exclusive and exhaustive.
- (a)
(i)
- (b)
(ii), (iv), (v)
- (c)
(vi), (iii)
- (d)
(i), (ii) and (iii)
Which of the following cannot be the probability of occurrence of an event?
- (a)
0
- (b)
\(\frac{1}{2}\)
- (c)
\(\frac{3}{4}\)
- (d)
\(\frac{4}{3}\)
From a set of 100 cards numbered 1 to 100, one card is drawn at random. The probability that the number obtained on the card is divisible by 6 or 8 but not by 24 is
- (a)
\(\frac{6}{25}\)
- (b)
\(\frac{1}{4}\)
- (c)
\(\frac{1}{6}\)
- (d)
\(\frac{1}{5}\)
If A and B are any two events having\(P(A\cup B)\frac { 1 }{ 2 } \) and \(P(\bar { A } )=\frac { 2 }{ 3 } \) then the probability of \(\bar { A } \cap B\) is
- (a)
\(\frac{1}{2}\)
- (b)
\(\frac{2}{3}\)
- (c)
\(\frac{1}{6}\)
- (d)
\(\frac{1}{3}\)
If A and B be two events associated with a random experiment such that P(A) = 0.3, P(B) = 0.2 and P(A\(\cap\)B) = 0.1, find P(\(\bar{A}\cap\)B)
- (a)
0.1
- (b)
0.2
- (c)
0.3
- (d)
0.4
Statement-I: The event E of a sample space 5 has occurred, if the outcome \(\omega \) of the experiment is such that \(\omega \epsilon \)E.
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement -1.
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement -1.
- (c)
If Statement -I is true but Statement-II is false.
- (d)
If Statement-I is false and Statement-II is true.
Statement-I: 5 = {I, 2, 3, 4, 5, 6}, then A: "a number less than or equal to 3 appears"
B : a number greater than or equal to 3 appears, then A and B are exhaustive events.
Statement-Il : Events are exhaustive if atleast one of them necessarily occur whenever the experiment is performed.
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement -1.
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct
explanation of Statement -1. - (c)
If Statement -I is true but Statement-II is false.
- (d)
If Statement-I is false and Statement-II is true.
In a non-leap year, the probability of having Tuesdays or 53 Wednesdays is
- (a)
\(\frac{1}{7}\)
- (b)
\(\frac{2}{7}\)
- (c)
\(\frac{3}{7}\)
- (d)
None of these
Select the CORRECT statement.
- (a)
If the probability that a person visiting a zoo will see the giratfee is 0.72 and the probability that he will see the bears is 0.84. Then the probability that he will see both is 0.52.
- (b)
The probabilities that a typist will make 0, 1,2, 3, 4, 5 or more mistakes in typing a report are, respectively, 0.12, 0.25,0.36,0.14,0.08 and 0.11.
- (c)
The probability of intersection of two events A and B is always less than or equal to those favourable to the event A.
- (d)
None of these.
Seven persons are to be seated in a row. The probability that two particular persons sit next to each other is
- (a)
\(\frac{1}{3}\)
- (b)
\(\frac{1}{6}\)
- (c)
\(\frac{2}{7}\)
- (d)
\(\frac{1}{2}\)
The probability that at least one of the events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.2, then P(\(\bar{A}\)) +P(\(\bar{B}\)) is
- (a)
0.4
- (b)
0.8
- (c)
1.2
- (d)
1.6