Question:
Let \(A\) and \(B\) be two events. Then \(1 + P(A \cap B) - P(B) - P(A)\) is equal to
Let \(A\) and \(B\) be two events. Then \(1 + P(A \cap B) - P(B) - P(A)\) is equal to
Show Hint
Expressions involving \(1 - P(\cdot)\) often represent complements — look for union or intersection identities.
Updated On: May 8, 2026
- \(P(\bar{A} \cup \bar{B})\)
- \(P(\bar{A} \cap \bar{B})\)
- \(P(\bar{A} \cap B)\)
- \(P(A \cup B)\)
- \(P(\bar{A} \cap \bar{B})\)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
Key identities:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
\[
P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B)
\]
Step 1: Start with given expression
\[ 1 + P(A \cap B) - P(A) - P(B) \]
Step 2: Rearrange terms
\[ = 1 - \left[ P(A) + P(B) - P(A \cap B) \right] \]
Step 3: Recognize identity
\[ P(A) + P(B) - P(A \cap B) = P(A \cup B) \]
Step 4: Substitute
\[ = 1 - P(A \cup B) \]
Step 5: Use complement rule
\[ 1 - P(A \cup B) = P(\bar{A} \cap \bar{B}) \]
Step 6: Final Answer
\[ \boxed{P(\bar{A} \cap \bar{B})} \]
Step 1: Start with given expression
\[ 1 + P(A \cap B) - P(A) - P(B) \]
Step 2: Rearrange terms
\[ = 1 - \left[ P(A) + P(B) - P(A \cap B) \right] \]
Step 3: Recognize identity
\[ P(A) + P(B) - P(A \cap B) = P(A \cup B) \]
Step 4: Substitute
\[ = 1 - P(A \cup B) \]
Step 5: Use complement rule
\[ 1 - P(A \cup B) = P(\bar{A} \cap \bar{B}) \]
Step 6: Final Answer
\[ \boxed{P(\bar{A} \cap \bar{B})} \]
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Probability Questions
- Let $A$ and $B$ be two events such that $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\right)P\left(B\right).$ If $0 < P\left(A\right)< 1$ and $0 < P\left(B\right)< 1$ , then $P\left(A\cup B\right)^{'}=$
- If $A$ and $B$ are mutually exclusive events and if $ p(B)=\frac{1}{3},p(A\cup B)=\frac{13}{21}, $ then $P(A)$ is equal to
- A complete cycle of a traffic light takes $60\, seconds$. During each cycle the light is green for $25\, seconds$, yellow for $5 \,seconds$ and red for $30\, seconds$. At a randomly chosen time, the probability that the light will not be green, is
- A die has four blank faces and two faces marked $3$. The chance of getting a total of $12$ in $5$ throws is
- Out of $15$ persons $10$ can speak Hindi and $8$ can speak English. If two persons are chosen at random, then the probability that one person speaks Hindi only and the other speaks both Hindi and English is
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions