ProbabilityHardBloom L3
Question
If $P(A) = x,P(B) = y$ and $x < y$, then
Options
A.$P(A/B) \in \left\lbrack \frac{x - 1}{y},\frac{x}{y} \right\rbrack$
B.$P(A/B) \in \left\lbrack \frac{x + y - 1}{y},\frac{x}{y} \right\rbrack$
C.$P(\text{ }A/B) \in \left\lbrack \frac{y - 1}{y},\frac{x}{y} \right\rbrack$
D.$P(A/B) \in \left\lbrack \frac{x}{y},\frac{x + 1}{y} \right\rbrack$
Solution
Sol. [Key Concept: Bound $P(A \cap \ B) \in \lbrack x + y - 1,x\rbrack$, then $P(A/B) = P(A \cap \ B)/y$.]
$$P(A \cap B) = P(A) + P(B) - P(A \cup B) \geq x + y - 1$$
$$\& P(\text{ }A \cap \text{ }B) \leq x$$
$$\Rightarrow \ P(A/B) = \frac{P(A \cap B)}{P(B)} \in \left\lbrack \frac{x + y - 1}{y},\frac{x}{y} \right\rbrack$$
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