GATE Data Science and Artificial Intelligence 2024 | Sample Paper | Question: 9

Question

For two events $\text{A}$ and $\text{B}$, $\text{B}$ $\subset$ $\text{A}$ Which of the following statement is correct?

• A. $P(B \mid A) \geq P(B)$ 
• B. $P(B \mid A) \leq P(B)$
• C. $P(A \mid B)<1$
• D. $P(A \mid B)=0$
   

   

Answer 1

Ans A:
Since B is proper subset of A, P(A$\cap$B) = P(B).

A: 𝑃(𝐵 | 𝐴) ≥ 𝑃(𝐵)

P(A$\cap$B)/ P(B) = 1

and 1  ≥ P(A)...(Basic probability rule).So option A is correct. Similarly we can check other options.

Answer 2

Answer : (A) 𝑃(𝐵 | 𝐴) ≥ 𝑃(𝐵)
Given $[B \subset A ]\rightarrow [P(A\cap B) = P(B) ]$   , Conditional Probability  :

so,

$P(\frac{A}{B})$ = $\frac{P(A \cap B)}{P(B)}$ = $\frac{P(B)}{P(B)} = 1$ 

and 

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

since $ 0 < P(B) \leq P(A) \leq 1 $,     

when some number P(B)  ( $\leq 1 )$ divided by another number greater than that number P(A)  ( $ P(B) \leq P(A) \leq 1 $ )

$\implies  \frac{P(B)}{P(A)} \geq P(B) \implies P(\frac{B} {A}) \geq P(B) $