Question

calculator
a and b are two events.
let ( p(a) = 0.3 ), ( p(b) = 0.8 ), and ( p(a \text{ and } b) = 0.24 )
which statement is true?

( circ ) a and b are not independent events because ( p(a|b)
eq p(a) ).
( circ ) a and b are independent events because ( p(a|b) = p(a) ) and ( p(b|a) = p(b) ).
( circ ) a and b are not independent events because ( p(a|b) = p(a) ) and ( p(b|a) = p(b) ).
( circ ) a and b are not independent events because ( p(a|b) = p(b) ) and ( p(b|a) = p(a) ).

Explanation:

Step1: Calculate P(A|B)

$P(A|B) = \frac{P(A \text{ and } B)}{P(B)} = \frac{0.24}{0.8} = 0.3$

Step2: Calculate P(B|A)

$P(B|A) = \frac{P(A \text{ and } B)}{P(A)} = \frac{0.24}{0.3} = 0.8$

Step3: Check independence

$P(A|B)=P(A)=0.3$ and $P(B|A)=P(B)=0.8$, so A and B are independent.

Answer:

B. A and B are independent events because $P(A|B) = P(A)$ and $P(B|A) = P(B)$