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

There are three bags B 1 , B 2 and B 3 . The bag B 1 contains 5 red and 5 green balls, B 2 contains 3 red and 5 green balls, and B 3 contains 5 red and 3 green balls, Bags B 1 , B 2 and B 3 have probabilities 3 10 , 3 10 and 4 10 respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

Probability that the chosen ball is green, given that the selected bag is B 3 , equals 3 8

Probability that the selected bag is B 3 and the chosen ball is green equals 3 10

Probability that the selected bag is B 3 , given that the chosen balls is green, equals 5 13

EASY

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Which of the following is not a property of a Binomial distribution?

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Important Questions on Probability

HARD

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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

HARD

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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

MEDIUM

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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

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

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

HARD

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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:

HARD

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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 :

MEDIUM

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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

HARD

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

Let two fair six-faced dice

A

and

B

be thrown simultaneously. If

E_{1}

is the event that die

A

shows up four,

E_{2}

is the event that die

B

shows up two and

E_{3}

is the event that the sum of numbers on both dice is odd, then which of the following statements is not true?

MEDIUM

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

If a fair coin is tossed

5

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

MEDIUM

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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

MEDIUM

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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:

HARD

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

An experiment succeeds twice as often as it fails. The probability of at least

5

successes in the six trials of this experiment is

HARD

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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,

HARD

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

Let

n_{1} & n_{2}

be the number of red and black balls, respectively, in box

I

. Let

n_{3} & n_{4}

be the number of red and black balls, respectively, in box

II

. One of the two boxes, box

I

and box

II

, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box

II

\frac{1}{3}

, then the correct option(s) with the possible values of

n_{1}, n_{2}, n_{3} & n_{4}

is(are)

HARD

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least

90 %

is:

HARD

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

There are three bags

B_{1}, B_{2}

and

B_{3} .

The bag

B_{1}

contains

5

red and

5

green balls,

B_{2}

contains

3

red and

5

green balls, and

B_{3}

contains

5

red and

3

green balls, Bags

B_{1}, B_{2}

and

B_{3}

have probabilities

\frac{3}{10}, \frac{3}{10}

and

\frac{4}{10}

respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

EASY

Mathematics>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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>Measures of Dispersion and Probability>Probability>Binomial and Poisson Distributions

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

Which of the following is not a property of a Binomial distribution?

Important Questions on Probability

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