A and B are two candidates seeking...

MathematicsAddition and Multiplication TheoremsJEE Advanced 1982Easy

$A$ and $B$ are two candidates seeking admission in IIT. The probability that $A$ is selected is 0.5 and the probability that both $A$ and $B$ are selected is almost 0.3. Is it possible that the probability of $B$ getting selected is 0.9?

Visualized Solution (English)

Visualizing the Sample Space

Let $S$ be the sample space of all outcomes.
Let $A$ be the event that candidate $A$ is selected.
Let $B$ be the event that candidate $B$ is selected.
Given: $P (A) = 0.5$ and $P (A \cap B) \leq 0.3$ .

The Addition Theorem

Using the Addition Theorem of Probability:
$P (A \cup B) = P (A) + P (B) - P (A \cap B)$

Applying the Probability Axiom

Since any probability is at most $1$ :
$P (A \cup B) \leq 1$
Substituting $P (A) = 0.5$ :
$0.5 + P (B) - P (A \cap B) \leq 1$

Isolating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P (B)

Rearranging the inequality to isolate $P (B)$ :
$P (B) \leq 1 - 0.5 + P (A \cap B)$
$P (B) \leq 0.5 + P (A \cap B)$

Using the Intersection Constraint

Given the constraint on the intersection:
$P (A \cap B) \leq 0.3$
Substituting this into our inequality:
$P (B) \leq 0.5 + 0.3$

The Final Verdict

Calculating the upper bound:
$P (B) \leq 0.8$
The question asks if $P (B) = 0.9$ is possible.
Since $0.9 > 0.8$ , it is not possible.

Key Takeaway & Extension

Key Takeaway: The probability of an event is constrained by $P (B) \leq 1 - P (A) + P (A \cap B)$ .
Next Challenge: What is the maximum value of $P (B)$ if $A$ and $B$ are independent events?

00:00 / 00:00

Conceptually Similar Problems

MathematicsConditional ProbabilityJEE Advanced 2003Moderate

View in:English Hindi

$A$ is targeting to $B, B$ and $C$ are targeting to $A$ . Probability of hitting the target by $A, B$ and $C$ are $2/3, 1/2$ and $1/3$ respectively. If $A$ is hit then find the probability that $B$ hits the target and $C$ does not.

Answer:

\frac{1}{2}

MathematicsClassical Definition of ProbabilityJEE Main 2005Easy

View in:English Hindi

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is

(A)

2/9

(B)

1/9

(C)

8/9

(D)

7/9

Answer:B

MathematicsAddition and Multiplication TheoremsJEE Main 2002Easy

View in:English Hindi

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

(B)

1/2

(C)

2/3

(D)

1/3

Answer:A

MathematicsAddition and Multiplication TheoremsJEE Advanced 1986Moderate

View in:English Hindi

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

Answer:C

MathematicsAddition and Multiplication TheoremsJEE Advanced 1980Easy

View in:English Hindi

Two events $A$ and $B$ have probabilities $0.25$ and $0.50$ respectively. The probability that both $A$ and $B$ occur simultaneously is $0.14$ . Then the probability that neither $A$ nor $B$ occurs is

(A)

0.39

(B)

0.25

(C)

0.11

(D)

none of these

Answer:A

MathematicsConditional ProbabilityJEE Advanced 1994Easy

View in:English Hindi

If two events $A$ and $B$ are such that $P (A^{c}) = 0.3, P (B) = 0.4$ and $P (A \cap B^{c}) = 0.5$ , then $P (B ∣ A \cup B^{c}) = .........$

Answer:1/4

MathematicsAddition and Multiplication TheoremsJEE Advanced 1987Easy

View in:English Hindi

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

(E)

none

Answer:C

MathematicsAddition and Multiplication TheoremsJEE Advanced 2004Moderate

View in:English Hindi

$A$ and $B$ are two independent events. $C$ is event in which exactly one of $A$ or $B$ occurs. Prove that $P (C) \geq P (A \cup B) P (\overset{ˉ}{A} \cap \overset{ˉ}{B})$ .

MathematicsAddition and Multiplication TheoremsJEE Advanced 2003Moderate

View in:English Hindi

For a student to qualify he must pass at least two out of three exams. The probability that he will pass the $1^{s t}$ exam is $p$ . If he fails in one of the exams then the probability of his passing in the next exam is $p /2$ otherwise it remains the same. Find the probability that he will qualify.

Answer:

2 p^{2} - p^{3}

MathematicsConditional ProbabilityJEE Advanced 1982Easy

View in:English Hindi

If $A$ and $B$ are two events such that $P (A) > 0$ , and $P (B) \neq = 1$ , then $P (\frac{A ˉ}{B ˉ})$ is equal to

(A)

1 - P (\frac{A}{B})

(B)

1 - P (\frac{A ˉ}{B})

(C)

\frac{1 - P ( A \cup B )}{P ( B ˉ )}

(D)

\frac{P ( A ˉ )}{P ( B ˉ )}

Answer:C

Visualized Solution (English)

Visualizing the Sample Space

The Addition Theorem

Applying the Probability Axiom

Isolating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P(B)

Using the Intersection Constraint

The Final Verdict

Key Takeaway & Extension

Conceptually Similar Problems

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If two events A and B are such that P(Ac)=0.3,P(B)=0.4 and P(A∩Bc)=0.5, then P(B∣A∪Bc)=.........

.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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A and B are two independent events. C is event in which exactly one of A or B occurs. Prove that P(C)≥P(A∪B)P(Aˉ∩Bˉ).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A and B are two events such that P(A)>0, and P(B)=1, then P(BˉAˉ​) is equal to

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If two events A and B are such that P(Ac)=0.3,P(B)=0.4 and P(A∩Bc)=0.5, then P(B∣A∪Bc)=.........

.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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A and B are two independent events. C is event in which exactly one of A or B occurs. Prove that P(C)≥P(A∪B)P(Aˉ∩Bˉ).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A and B are two events such that P(A)>0, and P(B)=1, then P(BˉAˉ​) is equal to

Isolating $P (B)$

If two events $A$ and $B$ are such that $P (A^{c}) = 0.3, P (B) = 0.4$ and $P (A \cap B^{c}) = 0.5$ , then $P (B ∣ A \cup B^{c}) = .........$

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

$A$ and $B$ are two independent events. $C$ is event in which exactly one of $A$ or $B$ occurs. Prove that $P (C) \geq P (A \cup B) P (\overset{ˉ}{A} \cap \overset{ˉ}{B})$ .

If $A$ and $B$ are two events such that $P (A) > 0$ , and $P (B) \neq = 1$ , then $P (\frac{A ˉ}{B ˉ})$ is equal to

If two events $A$ and $B$ are such that $P (A^{c}) = 0.3, P (B) = 0.4$ and $P (A \cap B^{c}) = 0.5$ , then $P (B ∣ A \cup B^{c}) = .........$

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

$A$ and $B$ are two independent events. $C$ is event in which exactly one of $A$ or $B$ occurs. Prove that $P (C) \geq P (A \cup B) P (\overset{ˉ}{A} \cap \overset{ˉ}{B})$ .

If $A$ and $B$ are two events such that $P (A) > 0$ , and $P (B) \neq = 1$ , then $P (\frac{A ˉ}{B ˉ})$ is equal to