Addition and Multiplication Theorems

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JEE Main 2016

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?

E_{1}

and

E_{3}

are independent

(B)

E_{1}, E_{2}

and

E_{3}

are independent

(C)

E_{1}

and

E_{2}

are independent

(D)

E_{2}

and

E_{3}

are independent

JEE Main 2014

LEVELJEE Main

Let $A$ and $B$ be two events such that $P (A \cup B) = 5/6, P (A \cap B) = 1/4$ and $P (\overset{ˉ}{A}) = 1/4$ , where $\overset{ˉ}{A}$ stands for the complement of the event $A$ . Then the events $A$ and $B$ are

(A)

independent but not equally likely

(B)

independent and equally likely

(C)

mutually exclusive and independent

(D)

equally likely but not independent

JEE Main 2007

LEVELBoard

Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is

The probability that $A$ speaks truth is $4/5$ , while the probability for $B$ is $3/4$ . The probability that they contradict each other when asked to speak on a fact is

Events $A, B, C$ are mutually exclusive events such that $P (A) = \frac{3 x + 1}{3}, P (B) = \frac{1 - x}{4}$ and $P (C) = \frac{1 - 2 x}{2}$ . The set of possible values of $x$ are in the interval.

(A)

[0, 1]

(B)

[1/3, 1/2]

(C)

[1/3, 2/3]

(D)

[1/13, 1/3]

JEE Main 2002

LEVELJEE Main

A problem in mathematics is given to three students $A, B, C$ and their respective probability of solving the problem is $1/2, 1/3$ and $1/4$ . Probability that the problem is solved is

(A)

3/4

$A$ and $B$ are events such that $P (A \cup B) = 3/4, P (A \cap B) = 1/4, P (\overset{ˉ}{A}) = 2/3$ then $P (\overset{ˉ}{A} \cap B)$ is

Four persons independently solve a certain problem correctly with probabilities $1/2, 3/4, 1/4, 1/8$ . Then the probability that the problem is solved correctly by at least one of them is

(A)

235/256

(B)

21/256

(C)

3/256

(D)

253/256

JEE Advanced 2011

LEVELJEE Main

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $11/25$ and the probability of none of them occurring is $2/25$ . If $P (T)$ denotes the probability of occurrence of the event $T$ , then

* Multiple Correct Options

(A)

P (E) = 4/5, P (F) = 3/5

(B)

P (E) = 1/5, P (F) = 2/5

(C)

P (E) = 2/5, P (F) = 1/5

(D)

P (E) = 3/5, P (F) = 4/5

JEE Advanced 2008

LEVELJEE Main

An experiment has 10 equally likely outcomes. Let $A$ and $B$ be non-empty events of the experiment. If $A$ consists of 4 outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is

A six faced fair dice is thrown until 1 comes, then the probability that 1 comes in even no. of trials is

If $P (B) = \frac{3}{4}, P (A \cap B \cap \overset{ˉ}{C}) = \frac{1}{3}$ and $P (\overset{ˉ}{A} \cap B \cap \overset{ˉ}{C}) = \frac{1}{3}$ , then $P (B \cap C)$ is

The probabilities that a student passes in Mathematics, Physics and Chemistry are $m, p$ and $c$ , respectively. Of these subjects, the student has a $75%$ chance of passing in at least one, a $50%$ chance of passing in at least two, and a $40%$ chance of passing in exactly two. Which of the following relations are true?

* Multiple Correct Options

(A)

p + m + c = 19/20

(B)

p + m + c = 27/20

(C)

p m c = 1/10

(D)

p m c = 1/4

JEE Advanced 1998

LEVELJEE Main

If from each of the three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, one ball is drawn at random, then the probability that 2 white and 1 black ball will be drawn is

(A)

13/32

(B)

1/4

(C)

1/32

(D)

3/16

JEE Advanced 1996

LEVELJEE Main

For the three events $A, B$ , and $C, P$ (exactly one of the events $A$ or $B$ occurs) $= P$ (exactly one of the events $B$ or $C$ occurs) $= P$ (exactly one of the events $C$ or $A$ occurs) $= p$ and $P$ (all the three events occur simultaneously) $= p^{2}$ , where $0 < p < 1/2$ . Then the probability of at least one of the three events $A, B$ and $C$ occurring is

(A)

\frac{3 p + 2 p ^{2}}{2}

(B)

\frac{p + 3 p ^{2}}{4}

(C)

\frac{p + 3 p ^{2}}{2}

(D)

\frac{3 p + 2 p ^{2}}{4}

JEE Advanced 1995

LEVELJEE Main

Let $0 < P (A) < 1, 0 < P (B) < 1$ and $P (A \cup B) = P (A) + P (B) - P (A) P (B)$ then

* Multiple Correct Options

(A)

P (B / A) = P (B) - P (A)

(B)

P (A^{c} - B^{c}) = P (A^{c}) - P (B^{c})

(C)

P (A \cup B)^{c} = P (A^{c}) P (B^{c})

(D)

P (A / B) = P (A)

JEE Advanced 1994

LEVELJEE Main

Let $A, B, C$ be three mutually independent events. Consider the two statements $S_{1}$ and $S_{2}$ \\ $S_{1} : A$ and $B \cup C$ are independent \\ $S_{2} : A$ and $B \cap C$ are independent \\ Then,

(A)

Both

S_{1}

and

S_{2}

are true

(B)

Only

S_{1}

is true

(C)

Only

S_{2}

is true

(D)

Neither

S_{1}

nor

S_{2}

is true

JEE Advanced 1993

LEVELBoard

An unbiased die with faces marked $1, 2, 3, 4, 5$ and $6$ is rolled four times. Out of four face values obtained, the probability that the minimum face value is not less than $2$ and the maximum face value is not greater than $5$ , is

(A)

16/81

(B)

1/81

(C)

80/81

(D)

65/81

JEE Advanced 1993

LEVELJEE Main

$E$ and $F$ are two independent events. The probability that both $E$ and $F$ happen is $1/12$ and the probability that neither $E$ nor $F$ happens is $1/2$ . Then,

* Multiple Correct Options

(A)

P (E) = 1/3, P (F) = 1/4

(B)

P (E) = 1/2, P (F) = 1/6

(C)

P (E) = 1/6, P (F) = 1/2

(D)

P (E) = 1/4, P (F) = 1/3

JEE Advanced 1992

LEVELBoard

India plays two matches each with West Indies and Australia. In any match the probabilities of India getting, points $0, 1$ and $2$ are $0.45, 0.05$ and $0.50$ respectively. Assuming that the outcomes are independent, the probability of India getting at least $7$ points is

(A)

0.8750

(B)

0.0875

(C)

0.0625

(D)

0.0250

JEE Advanced 1991

LEVELJEE Main

For any two events $A$ and $B$ in a sample space

* Multiple Correct Options

(A)

P (A / B) \geq \frac{P ( A ) + P ( B ) - 1}{P ( B )}, P (B) \neq = 0

is always true

(B)

P (A \cap \overset{ˉ}{B}) = P (A) - P (A \cap B)

does not hold

(C)

P (A \cup B) = 1 - P (\overset{ˉ}{A}) P (\overset{ˉ}{B})

, if

A

and

B

are independent

(D)

P (A \cup B) = 1 - P (\overset{ˉ}{A}) P (\overset{ˉ}{B})

, if

A

and

B

are disjoint.

JEE Advanced 1989

LEVELJEE Main

If $E$ and $F$ are independent events such that $0 < P (E) < 1$ and $0 < P (F) < 1$ , then

* Multiple Correct Options

(A)

E

and

F

are mutually exclusive

(B)

E

and

F^{c}

(the complement of the event

F

) are independent

(C)

E^{c}

and

F^{c}

are independent

(D)

P (E ∣ F) + P (E^{c} ∣ F) = 1

JEE Advanced 1988

LEVELBoard

For two given events $A$ and $B, P (A \cap B)$ is

* Multiple Correct Options

(A)

not less than

P (A) + P (B) - 1

(B)

not greater than

P (A) + P (B)

(C)

equal to

P (A) + P (B) - P (A \cup B)

(D)

equal to

P (A) + P (B) + P (A \cup B)

JEE Advanced 1987

LEVELBoard

The probability that at least one of the events $A$ and $B$ occurs is $0.6$ . If $A$ and $B$ occur simultaneously with probability $0.2$ , then $P (\overset{ˉ}{A}) + P (\overset{ˉ}{B})$ is

(A)

0.4

(B)

0.8

(C)

1.2

(D)

1.4

A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II and III are $p, q$ and $\frac{1}{2}$ respectively. If the probability that the student is successful is $\frac{1}{2}$ , then

(A)

p = q = 1

(B)

p = q = \frac{1}{2}

(C)

p = 1, q = 0

(D)

p = 1, q = \frac{1}{2}

(E)

none of these

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A and B be two events such that P(A∪B)=5/6,P(A∩B)=1/4 and P(Aˉ)=1/4, where Aˉ stands for the complement of the event A. Then the events A and B are

Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability that A speaks truth is 4/5, while the probability for B is 3/4. The probability that they contradict each other when asked to speak on a fact is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Events A,B,C are mutually exclusive events such that P(A)=33x+1​,P(B)=41−x​ and P(C)=21−2x​. The set of possible values of x are in the interval.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A and B are events such that P(A∪B)=3/4,P(A∩B)=1/4,P(Aˉ)=2/3 then P(Aˉ∩B) is

A six faced fair dice is thrown until 1 comes, then the probability that 1 comes in even no. of trials is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P(B)=43​,P(A∩B∩Cˉ)=31​ and P(Aˉ∩B∩Cˉ)=31​, then P(B∩C) is

If from each of the three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, one ball is drawn at random, then the probability that 2 white and 1 black ball will be drawn is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let 0<P(A)<1,0<P(B)<1 and P(A∪B)=P(A)+P(B)−P(A)P(B) then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A,B,C be three mutually independent events. Consider the two statements S1​ and S2​ \\ S1​:A and B∪C are independent \\ S2​:A and B∩C are independent \\ Then,

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } E and F are two independent events. The probability that both E and F happen is 1/12 and the probability that neither E nor F happens is 1/2. Then,

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For any two events A and B in a sample space

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If E and F are independent events such that 0<P(E)<1 and 0<P(F)<1, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For two given events A and B,P(A∩B) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability that at least one of the events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.2, then P(Aˉ)+P(Bˉ) is

Let $A$ and $B$ be two events such that $P (A \cup B) = 5/6, P (A \cap B) = 1/4$ and $P (\overset{ˉ}{A}) = 1/4$ , where $\overset{ˉ}{A}$ stands for the complement of the event $A$ . Then the events $A$ and $B$ are

The probability that $A$ speaks truth is $4/5$ , while the probability for $B$ is $3/4$ . The probability that they contradict each other when asked to speak on a fact is

Events $A, B, C$ are mutually exclusive events such that $P (A) = \frac{3 x + 1}{3}, P (B) = \frac{1 - x}{4}$ and $P (C) = \frac{1 - 2 x}{2}$ . The set of possible values of $x$ are in the interval.

$A$ and $B$ are events such that $P (A \cup B) = 3/4, P (A \cap B) = 1/4, P (\overset{ˉ}{A}) = 2/3$ then $P (\overset{ˉ}{A} \cap B)$ is

If $P (B) = \frac{3}{4}, P (A \cap B \cap \overset{ˉ}{C}) = \frac{1}{3}$ and $P (\overset{ˉ}{A} \cap B \cap \overset{ˉ}{C}) = \frac{1}{3}$ , then $P (B \cap C)$ is

Let $0 < P (A) < 1, 0 < P (B) < 1$ and $P (A \cup B) = P (A) + P (B) - P (A) P (B)$ then

Let $A, B, C$ be three mutually independent events. Consider the two statements $S_{1}$ and $S_{2}$ \\ $S_{1} : A$ and $B \cup C$ are independent \\ $S_{2} : A$ and $B \cap C$ are independent \\ Then,

$E$ and $F$ are two independent events. The probability that both $E$ and $F$ happen is $1/12$ and the probability that neither $E$ nor $F$ happens is $1/2$ . Then,

For any two events $A$ and $B$ in a sample space

If $E$ and $F$ are independent events such that $0 < P (E) < 1$ and $0 < P (F) < 1$ , then

For two given events $A$ and $B, P (A \cap B)$ is

The probability that at least one of the events $A$ and $B$ occurs is $0.6$ . If $A$ and $B$ occur simultaneously with probability $0.2$ , then $P (\overset{ˉ}{A}) + P (\overset{ˉ}{B})$ is