Explanation: The given condition implies that events and are independent . 1. Property of Independent Events: If two events and are independent, then their complements and are also independent. This means: 2. Applying De Morgan's Law: According to De Morgan's Law in set theory and probability: Therefore, the probability can be written as: 3. Combining the results: Since and are independent (as established in Step 1): Substituting this back into the De Morgan's equation: 4. Why other options are incorrect: For A and D: If and are independent, then and . The subtraction of probabilities shown in the options is not a standard property of independence. For B: This is not a standard identity for independent events. Conclusion: Option C is a direct consequence of the independence of events and . Correct Opti

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NIMCET 2022 — Mathematics PYQ

NIMCET | Mathematics | 2022

If $0<P(A)<1$ and $0<P(B)<1$ , and $P (A \cap B) = P (A) P (B)$ , then

Choose the correct answer:

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

C.
$P ((A \cup B)^{c}) = P (A^{c}) P (B^{c})$
(Correct Answer)

D.
$P (A ∣ B) = P (A) - P (B)$

Explanation

The given condition $P(A \cap B) = P(A)P(B)$ implies that events $A$ and $B$ are independent.

1. Property of Independent Events:

If two events $A$ and $B$ are independent, then their complements $A^c$ and $B^c$ are also independent. This means:

$P(A^c \cap B^c) = P(A^c)P(B^c)$

2. Applying De Morgan's Law:

According to De Morgan's Law in set theory and probability:

$(A \cup B)^c = A^c \cap B^c$

Therefore, the probability can be written as:

$P((A \cup B)^c) = P(A^c \cap B^c)$

3. Combining the results:

Since $A^c$ and $B^c$ are independent (as established in Step 1):

$P(A^c \cap B^c) = P(A^c)P(B^c)$

Substituting this back into the De Morgan's equation:

$P((A \cup B)^c) = P(A^c)P(B^c)$

4. Why other options are incorrect:

For A and D: If $A$ and $B$ are independent, then $P(B|A) = P(B)$ and $P(A|B) = P(A)$ . The subtraction of probabilities shown in the options is not a standard property of independence.

For B: This is not a standard identity for independent events.

Conclusion:

Option C is a direct consequence of the independence of events $A$ and $B$ .

Correct Option:

C) $P((A \cup B)^c) = P(A^c)P(B^c)$

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