IISER Mathematics - Probability
Exam Duration: 45 Mins Total Questions : 30
The probability that a teacher will give an unannounced test during any class is \(\cfrac { 1 }{ 5 } \). If a student is absent twice, the probability that he misses at least one test, is
- (a)
\(\cfrac { 16}{ 25 } \)
- (b)
\(\cfrac { 9 }{ 25 } \)
- (c)
\(\cfrac { 1 }{ 25 } \)
- (d)
NONE OF THESE
a bag contains five white and three black balls. Four balls are drawn one by one without replacement. The probability that the balls are alternately of different colours, is
- (a)
\(\cfrac { 1 }{ 4 } \)
- (b)
\(\cfrac { 1 }{ 6 } \)
- (c)
\(\cfrac { 1 }{ 7 } \)
- (d)
\(\cfrac { 1 }{ 8 } \)
Out of n persons sitting in a row, the probability that two of the selected persons do not sit side by side, is
- (a)
\(\cfrac { 2 }{ n } \)
- (b)
\(1-\cfrac { 1 }{ n } \)
- (c)
\(1-\cfrac { 2 }{ n } \)
- (d)
None of these
If p is the probability that a man aged 80 years will die in a year. Then the probability that out of n such men A1, A2, A3,....,An, each aged 80 years at least one dies in a year and it is the A1 to die first, is
- (a)
\(\cfrac { 1 }{ n } \{ 1-{ (1-p) }^{ n }\} \)
- (b)
\(\cfrac { 1 }{ n } { (1-p) }^{ n }\)
- (c)
\(\cfrac { { p }^{ n } }{ n } \)
- (d)
\(\cfrac { 1 }{ n } \)
The probability that atleast one of the events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.2, then \(P(\overline { A } )+P(\overline { B } )\) is
- (a)
0.4
- (b)
0.8
- (c)
1.2
- (d)
1.4
A and B throw with three dice and if A throws 8, the chance of B throwing a higher number is
- (a)
\(\cfrac { 7 }{27 } \)
- (b)
\(\cfrac { 20 }{27 } \)
- (c)
\(\cfrac { 13 }{27 } \)
- (d)
NONE OF THESE
A coin is tossed 2n times. The chance that number of times one gets head is not equal to the number of times one gets tail is
- (a)
\(\cfrac { (2n)! }{ (n!) } { \left( \cfrac { 1 }{ 2 } \right) }^{ 2n }\)
- (b)
\(1-\cfrac { (2n)! }{ (n!) } \)
- (c)
\(1-\cfrac { (2n)! }{ (n!) } { \left( \cfrac { 1 }{ { 4 }^{ n } } \right) }\)
- (d)
NONE OF THESE
If A and B are two events such that \(P(A)={1\over2}\), \(P(B)={1\over3}\), \(P({A\over B})={1\over4}\) then \(P(A'\cap B')\) is equal to
- (a)
1/12
- (b)
3/4
- (c)
1/4
- (d)
3/16
Let 0<P(A)<1, 0<P(B)<1 and P(\(A\cup B\))=P(A)+P(B)-P(A).P(B). Then,
- (a)
\(P({B\over A})=P(B)-P(A)\)
- (b)
\(P(A^c\cup B^c)=P(A^c)+P(B^c)\)
- (c)
\(P(A\cup B)^c=P(A^c).P(B^c)\)
- (d)
\(P({A\over B})={P({B\over A})}\)
A die is thrown. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then, \(P(A\cup B)\) is
- (a)
\(2\over5\)
- (b)
\(3\over5\)
- (c)
0
- (d)
1
The decimal parts of the logarithms of two numbers taken at random are found to six places. Probability that second can be subtracted from one first without borrowing is
- (a)
\(\left( \frac { 1 }{ 2 } \right) ^{ 6 }\)
- (b)
\(\left( \frac { 9 }{ 20 } \right) ^{ 6 }\)
- (c)
\(\left( \frac { 11 }{ 20 } \right) ^{ 6 }\)
- (d)
\(\left( \frac { 3 }{ 20 } \right) ^{ 6 }\)
Two dice are rolled one after another. The probability that the number on the first is less than or equal to the number on the second is
- (a)
\(\frac { 5 }{ 12 } \)
- (b)
\(\frac { 7 }{ 12 } \)
- (c)
\(\frac { 5 }{ 18 } \)
- (d)
\(\frac { 13 }{ 18 } \)
A sum of money is rounded off to the nearest rupee, the probability that the rounded off error is atleast ten paise is
- (a)
\(\frac { 1 }{ 100 } \)
- (b)
\(\frac { 9 }{ 100 } \)
- (c)
\(\frac { 49 }{ 100 } \)
- (d)
\(\frac { 81 }{ 100 } \)
If the probability for A to fail in an examination is 0.2 and that for B is 0.3, then the probability that either A or B fails,
- (a)
0.38
- (b)
0.44
- (c)
0.50
- (d)
0.94
Eight coins are tossed at a time, the probability of getting atleast 6 heads up, is
- (a)
\(\frac { 7 }{ 64 } \)
- (b)
\(\frac { 57 }{ 64 } \)
- (c)
\(\frac { 37 }{ 256 } \)
- (d)
\(\frac { 229 }{ 256 } \)
Two numbers x and y are chosen at random from the set {1, 2, 3, ... , 30}. The probability that x 2 - y2 is divisible by 3 is
- (a)
3/29
- (b)
4/29
- (c)
5/29
- (d)
none of these
Two distinct numbers are selected at random from the first twelve natural numbers. The probability that the sum will be divisible by 3 is
- (a)
1/3
- (b)
23/66
- (c)
1/2
- (d)
none of these
Sixteen players P1 , P2, ...P16 play in a tournament. They are divided into eight pairs at random. From each pair a winner is decided on the basis of a game played between the two players of the pair. Assuming that all the players are of equal strength, the probability that exactly one of the two players P1 and P2 is among the eight winners is
- (a)
4/15
- (b)
7/15
- (c)
8/15
- (d)
17/30
If A and B are two events such that P(A) = 1/2 and P(B) = 2/3, then
- (a)
\(P(A\cup B)\ge 2/3\)
- (b)
\(P(A\cap { B }^{ ' })\le 1/3\)
- (c)
\(1/6\le P(A\cap B)\le 1/2\)
- (d)
\(1/6\le P(A^{ ' }\cap B)\le 1/2\)
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)
\(E(X)=\frac { 25 }{ 32 } \)
The total number of elementary events associated to the random experiment of throwing three die together is
- (a)
210
- (b)
216
- (c)
215
- (d)
220
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
An experiment involves rolling a pair of dice. The following events are recorded.
P: The sum is greater than 9.
Q: 1 occurs on either die.
R: The sum is atleast 8 and a multiple of 3.
Which pair of these events is/are mutually exclusive?
- (a)
P and Q
- (b)
Q and R
- (c)
Both (a) and (b)
- (d)
None of these
Out of 20 positive consecutive integers, two are chosen at random. The probability that their sum is odd, is
- (a)
\(\frac{1}{20}\)
- (b)
\(\frac{19}{20}\)
- (c)
\(\frac{10}{19}\)
- (d)
\(\frac{9}{19}\)
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 3 or 4 but not by 12 is
- (a)
\(\frac{1}{2}\)
- (b)
\(\frac{1}{4}\)
- (c)
\(\frac{1}{6}\)
- (d)
\(\frac{2}{5}\)
A and B are two events such that P(A)=0.54, P(B)=0.69 and P(A\(\cap\)B)=0.35. Find (i) P(A'\(\cap\)B') (ii) P(A\(\cap\)B')
- (a)
(i) (ii) 0.12 0.19 - (b)
(i) (ii) 0.19 0.12 - (c)
(i) (ii) 0.13 0.20 - (d)
(i) (ii) 0.19 0.18
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.
Statement-I : Events E and F are such that P(not Eor not F) = 0.25, then E and F are not mutually exclusive.
Statement-II: A and B are events such that P(A) = 0.42,P(B) =0.48 andP(A andB) = 0.16 then P(notA), P(not B) and P(A or B) are 0,58, 0.52 and 0.74, respectively .
- (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.
Three numbers are chosen from 1 to 20. Find the probability that they are not consecutive.
- (a)
\(\frac{186}{190}\)
- (b)
\(\frac{187}{190}\)
- (c)
\(\frac{188}{190}\)
- (d)
\(\frac{18}{^{20}C_{3}}\)
6 boys and 6 girls sit in a row at random. The probability that all the girls sit together is
- (a)
\(\frac{1}{432}\)
- (b)
\(\frac{21}{431}\)
- (c)
\(\frac{1}{132}\)
- (d)
\(\frac{2}{132}\)