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Let $S$ be a set with $3$ elements. What is the probability of choosing an ordered pair $(A, B)$ of subsets of $S$ such that $A$ and $B$ are disjoint?

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Important Questions on Probability

MEDIUM

Mathematics>Statistics and Probability>Probability>Basics of Probability

A letter is taken at random from the word "STATISTICS" and another letter is taken at random from the word "ASSISTANT". The probability that they are same letters is

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

If $12$ identical balls are to be placed in $3$ identical boxes, then the probability that one of the boxes contains exactly $3$ balls is

MEDIUM

Mathematics>Statistics and Probability>Probability>Basics of Probability

Let

a

and

b

2

consecutive integers selected from the first

20

natural numbers. The probability that

\sqrt{a^{2} + b^{2} + a^{2} b^{2}}

is an odd positive integer is

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

Let

X_{n} = \{1, 2, 3, \dots ., n\}

and let a subset

A

X_{n}

be chosen so that every pair of elements of A differ by at least

3

. (For example, if

n = 5,

A can be

\emptyset, \{2\} o r \{1, 5\}

Among others). When

n = 10,

let the probability that

1 \in A

p

and let the probability that

2 \in A

q .

Then-

EASY

Mathematics>Statistics and Probability>Probability>Basics of Probability

The probability that among

7

persons, no

2

were born on the same day of a week is

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

From a group of

10

men and

5

women, four member committees are to be formed each of which must contain at least one women. Then the probability for these committees to have more women than men, is :

EASY

Mathematics>Statistics and Probability>Probability>Basics of Probability

An ellipse of eccentricity

\frac{2 \sqrt{2}}{3}

is inscribed in a circle. A point is chosen inside the circle at random. The probability that the point lies outside the ellipse is

MEDIUM

Mathematics>Statistics and Probability>Probability>Basics of Probability

A box contains coupons labelled

1, 2, . . . ., 100

. Five coupons are picked at random one after another without replacement. Let the numbers on the coupons be

x_{1}, x_{2}, \dots ., x_{5}

. What is the probability that

x_{1} > x_{2} > x_{3} a n d x_{3} < x_{4} < x_{5}

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

Two different families

A

and

B

are blessed with equal number of children. There are

3

tickets to be distributed amongst the children of these families so that no child gets more than one ticket. If the probability that all the tickets go to the children of the family

B

\frac{1}{12},

then the number of children in each family is

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

If two different numbers are taken from the set

\{0, 1, 2, 3, . . . . ., 10\}

; then the probability that their sum as well as absolute difference are both multiple of

4

, is:

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

There are 6 boxes labelled

B_{1}, B_{2}, \dots ., B_{6} .

In each trial, two fair dice

D_{1}, D_{2}

are thrown. If

D_{1}

shows j and

D_{2}

shows

k

, then

j

balls are put into the box

B_{k} .

After

n

trials, what is the probability that

B_{1}

contains at most one ball?

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

Let

X

be a set containing

10

elements and

P (X)

be its power set. If

A

and

B

are picked up at random from

P (X)

, with replacement, then the probability that

A

and

B

have equal number of elements is:

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

An urn contains marbles of four colours : red, white, blue and green. When four marbles are drawn without replacement, the following events are equally likely:

(i) The selection of four red marbles

(ii) The selection of one white and three red marbles

(iii) The selection of one white, one blue and two red marbles

(iv) The selection of one marble of each colour

The smallest total number of marbles satisfying the given condition is

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

If a

3 -

digit number is randomly chosen. What is the probability that either the number itself or some permutation of the number (which is a

3 -

digit number) is divisible by

4

and

5 ?

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

A box contains coupons labeled 1, 2, 3 .....

n

. A coupon is picked at random and the number

x

is noted. The coupon is put back into the box and a new coupon is picked at random. The number is

y

. Then the probability that one of the numbers

x, y

divides the other is (in the options below

[r]

denotes the largest integer less than or equal to

r

)

MEDIUM

Mathematics>Statistics and Probability>Probability>Basics of Probability

A determinant of second order is made with the elements 0, 1. What is the probability that the determinant is positive?

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

Choose a number

n

uniformly at random from the set

\{1, 2, 3, . . . . ., 99, 100\}

. Choose one of the first seven days of the year

2014

at random and consider

n

consecutive days starting from the chosen day. What is the probability that among the chosen

n

days, the number of Sundays is different from the number of Mondays?

MEDIUM

Mathematics>Statistics and Probability>Probability>Basics of Probability

A quadratic equation

a x^{2} + b x + c = 0

with distinct coefficients is formed. It

a, b, c

are chosen from the numbers

2, 3

and

5,

then the probability that the equation has real roots is

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

If the lengths of the sides of a triangle are decided by the three throws of a single fair die, then the probability that the triangle is of maximum area given that it is an isosceles triangle, is:

HARD

Mathematics>Statistics and Probability>Probability>Basics of Probability

A multiple choice examination has

5

questions. Each question has three alternative answers out of which exactly one is correct. The probability that a student will get

4

or more correct answers just by guessing is :

Let S be a set with 3 elements. What is the probability of choosing an ordered pair (A, B) of subsets of S such that A and B are disjoint?

Important Questions on Probability

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Let $S$ be a set with $3$ elements. What is the probability of choosing an ordered pair $(A, B)$ of subsets of $S$ such that $A$ and $B$ are disjoint?