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

Let A and B be two events such that P A ∪ B ¯ = 1 6 , P A ∩ B = 1 4 and P A ¯ = 1 4 , where A ¯ stands for the complement of the event A . Then the events A and B are

Independent but not equally likely.

Independent and equally likely.

Mutually exclusive and independent.

Equally likely but not independent.

EASY

Earn 100

A class of undergraduate students were asked in $2016$ their major subject and whether they listen to music on their commute to university. $S$ is the set of science majors and $M$ is the set of students who listen to music on their commute.

Find $P (S) \times P (M)$ and $P (S \cap M)$ , and hence determine whether $S$ and $M$ are independent events, stating a reason for your answer.

Important Questions on Probability

HARD

Mathematics>Algebra>Probability>Conditional Probability

Let

E & F

be two independent events. The probability that

E & F

happen is

\frac{1}{12}

and the probability that neither

E

nor

F

happens is

\frac{1}{2}

, then a value of

\frac{P (E)}{P (F)}

is:

MEDIUM

Mathematics>Algebra>Probability>Conditional Probability

In a class,

40

% of students study Maths and Science and

60

% of students study Maths. What is the probability of a students studying Science given the student is already studying Maths?

EASY

Mathematics>Algebra>Probability>Conditional Probability

A

and

B

are two events such that

P (A) \neq 0

and

P (B| A) = 1

, then

HARD

Mathematics>Algebra>Probability>Conditional Probability

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

EASY

Mathematics>Algebra>Probability>Conditional Probability

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>Algebra>Probability>Conditional Probability

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) .

EASY

Mathematics>Algebra>Probability>Conditional Probability

A

and

B

are any two events such that

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

, then

HARD

Mathematics>Algebra>Probability>Conditional Probability

Let

|X|

denote the number of elements in set

X .

Let

S = \{1, 2, 3, 4, 5, 6\}

be a sample space, where each element is equally likely to occur. If

A

and

B

are independent events associated with

S,

then the number of ordered pairs

(A, B)

such that

1 \leq |B| < |A|,

equals

HARD

Mathematics>Algebra>Probability>Conditional Probability

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>Algebra>Probability>Conditional Probability

The probability that a person will develop immunity after vaccination is

0.8.

8

people are given the vaccine then probability that all develop immunity is equal to

HARD

Mathematics>Algebra>Probability>Conditional Probability

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?

EASY

Mathematics>Algebra>Probability>Conditional Probability

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>Algebra>Probability>Conditional Probability

A

and

B

are any two events such that

P (A) = \frac{2}{5}

and

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

then the conditional probability,

P (A| (A^{'} \cup B^{'})),

where

A^{'}

denotes the complement of

A

, is equal to :

HARD

Mathematics>Algebra>Probability>Conditional Probability

Let

A

and

B

be two events such that

P (\bar{A \cup B}) = \frac{1}{6}, P (A \cap B) = \frac{1}{4}

and

P (\bar{A}) = \frac{1}{4},

where

\bar{A}

stands for the complement of the event

A

. Then the events

A

and

B

are

MEDIUM

Mathematics>Algebra>Probability>Conditional Probability

A problem in mathematics is given to three students

A, B, C

and their respective probability of solving the problem is

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

and

\frac{1}{4}

, the probability that the problem will be solved is

MEDIUM

Mathematics>Algebra>Probability>Conditional Probability

Four persons can hit a target correctly with probabilities

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

and

\frac{1}{8}

respectively. If all hit at the target independently, then the probability that the target would be hit, is

MEDIUM

Mathematics>Algebra>Probability>Conditional Probability

Let

A

and

B

be two non-null events such that

A \subset B .

Then, which of the following statements is always correct?

EASY

Mathematics>Algebra>Probability>Conditional Probability

A coin is tossed and a die is rolled. The probability that the coin shows head and the die shows

3

MEDIUM

Mathematics>Algebra>Probability>Conditional Probability

If a fair coin is tossed

5

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

HARD

Mathematics>Algebra>Probability>Conditional Probability

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 :

Important Questions on Probability

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