Let A and E be any two events with positive probabilities Statement I: P E / A ≥ P A / E P E . Statement II: P A / E ≥ P A ∩ E .

Statement - I is false, Statement - II is true

Statement - I is true, Statement - II is false

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From a bag containing $100$ tickets numbered $1, 2, 3, . . . . . . . . . . ., 100$ one ticket is drawn. If the number on this ticket is $x$ , then the probability that $x + \frac{1}{x} > 2$ is_____

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Important Questions on Probability

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Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

A box contains

b

blue balls and

r

red balls. A ball is drawn randomly from the box and is returned to the box with another ball of the same colour. The probability that the second ball drawn from the box is blue, is

EASY

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

A

and

B

are events with

P (A \cup B) = \frac{3}{4}, P (A^{'}) = \frac{2}{3}

and

P (A \cap B) = \frac{1}{4}

then

P (B)

HARD

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

Let

A, B

and

C

be three events, which are pair-wise independent and

\bar{E}

denotes the complement of an event

E

. If

P (A \cap B \cap C) = 0

and

P (C) > 0,

then

P [(\bar{A} \cap \bar{B}) | C]

is equal to

EASY

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

A

and

B

are any two events such that

P (A) + P (B) - P (A a n d B) = P (A)

, then

EASY

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

An urn contains

5

red and

2

green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is:

HARD

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

Four persons independently solve a certain problem correctly with probabilities

\frac{1}{2}, \frac{3}{4}, \frac{1}{4}, \frac{1}{8}

. Then the probability that the problem is solved correctly by at least one of them is

EASY

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

For the events

A

and

B,

P (A) = \frac{3}{4}, P (B) = \frac{1}{5}, P (A \cap B) = \frac{1}{20}

then

P (\frac{A}{B}) =

HARD

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

Let

A

and

E

be any two events with positive probabilities

Statement I:

P (E / A) \geq P (A / E) P (E) .

Statement II:

P (A / E) \geq P (A \cap E) .

HARD

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

If the mean and the variance of a binomial variate

X

are

2 & 1

respectively, then the probability that

X

takes a value greater than or equal to one is:

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Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

If a fair coin is tossed

5

times, the probability that heads does not occur two or more times in a row is

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Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

A

and

B

are two events such that

P (A) \neq 0

and

P (B| A) = 1

, then

HARD

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered

1, 2, 3, \dots, 9

is randomly picked and the number on the card is noted. The probability that the noted number is either

7

8

HARD

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

Two persons

A

and

B

throw a (fair) die

(

six-faced cube with faces numbered from

1

6

)

alternately, starting with

A

. The first person to get an outcome different from the previous one by the opponent wins.The probability that

B

wins is,

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Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

For the following distribution function $F (x)$ of a random variable $X$

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$F (x)$	$0.2$	$0.37$	$0.48$	$0.62$	$0.85$	$1$

$P (3 < X \leq 5) =$

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Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

A

and

B

are two events such that

P (A \cup B) = P (A \cap B)

, then the incorrect statement amongst the following statements is :

HARD

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

One ticket is selected at random from

50

tickets numbered

00, 01, 02, . . . ., 49

. Then, the probability that the sum of the digits on the selected ticket is

8

, given that the product of these digits is zero, equals

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Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

A bag contains

4

red and

6

black balls. A ball is drawn at random from the bag, its color is observed and this ball along with two additional balls of the same color are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is:

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Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

A shooter can hit a given target with probability

\frac{1}{4} .

She keeps firing a bullet at the target until she hits it successfully three times, and then she stops firing. The probability that she fires exactly six bullets lies in the interval

HARD

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to :

HARD

Mathematics>Statistics and Probability>Probability>Probability - A Theoretical Approach

A computer producing factory has only two plants $T_{1}$ and $T_{2}$ . Plant $T_{1}$ produces $20 %$ and plant $T_{2}$ produces $80 %$ of the total computers produced. $7 %$ of computers produced in the factory turn out to be defective. The probability that a computer turns out to be defective which is produced in plant $T_{1}$ is ten times of the computers produced in the plant $T_{2}$ . A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $T_{2}$ is

From a bag containing 100 tickets numbered 1, 2, 3, ...........,100 one ticket is drawn. If the number on this ticket is x, then the probability that x+1x>2 is_____

Important Questions on Probability

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From a bag containing $100$ tickets numbered $1, 2, 3, . . . . . . . . . . ., 100$ one ticket is drawn. If the number on this ticket is $x$ , then the probability that $x + \frac{1}{x} > 2$ is_____