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JAMIA 2022 — Mathematics PYQ

JAMIA | Mathematics | 2022

If A and B are two independent events in sample space, then $P (\overset{ˉ}{A} ∣ \overset{ˉ}{B}) equals:$

Choose the correct answer:

B.
$; 1 - P (\overset{ˉ}{A} ∣ \overset{ˉ}{B})$

C.
$; 1 - P (B)$

D.
$; 1 - P (A)$
(Correct Answer)

Explanation

1. Definition of Conditional Probability

The formula for conditional probability of an event $X$ given $Y$ is:

$P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)}$

Applying this to our specific case:

$P(\bar A \mid \bar B) = \frac{P(\bar A \cap \bar B)}{P(\bar B)}$

2. Property of Independent Events

If $A$ and $B$ are independent events, then their complements $\bar A$ and $\bar B$ are also independent. This is a fundamental theorem in probability.

Because they are independent, the probability of their intersection is the product of their individual probabilities:

$P(\bar A \cap \bar B) = P(\bar A) \cdot P(\bar B)$

3. Substitution and Simplification

Now, substitute the intersection from Step 2 back into the formula from Step 1:

$P(\bar A \mid \bar B) = \frac{P(\bar A) \cdot P(\bar B)}{P(\bar B)}$

The $P(\bar B)$ terms in the numerator and denominator cancel out (assuming $P(\bar B) \neq 0$ ):

$P(\bar A \mid \bar B) = P(\bar A)$

4. Alternative Forms

Since $P(\bar A) = 1 - P(A)$ , the answer can also be written as:

$P(\bar A \mid \bar B) = 1 - P(A)$

Choose the correct answer:

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