Question

confirming independence
$p(a\cap b)=p(a)\cdot p(b)$
this equation is known as necessary and sufficient. it works exactly like a biconditional statement: two events a and b are independent if and only if the equation is true. it is a must!
based upon the definition of independence, determine if each set of events below are independent.
a) $p(a)=0.45$ $p(b)=0.30$ $p(a\cap b)=0.75$ choose your answer...

b) $p(a)=0.12$ $p(b)=0.56$ $p(a\cap b)=0.0672$ choose your answer...

c) $p(a)=\frac{4}{5}$ $p(b)=\frac{3}{8}$ $p(a\cap b)=\frac{7}{40}$ choose your answer...

d) $p(a)=\frac{7}{9}$ $p(b)=\frac{3}{4}$ $p(a\cap b)=\frac{7}{12}$ choose your answer...

Explanation:

Step1: Recall Independence Formula

For independent events, \( P(A \cap B) = P(A) \cdot P(B) \). We'll check each part by calculating \( P(A) \cdot P(B) \) and comparing to \( P(A \cap B) \).

Step2: Part A

Calculate \( P(A) \cdot P(B) = 0.45 \cdot 0.30 = 0.135 \). Given \( P(A \cap B) = 0.75 \). Since \( 0.135
eq 0.75 \), events are not independent.

Step3: Part B

Calculate \( P(A) \cdot P(B) = 0.12 \cdot 0.56 = 0.0672 \). Given \( P(A \cap B) = 0.0672 \). Since \( 0.0672 = 0.0672 \), events are independent.

Step4: Part C

Calculate \( P(A) \cdot P(B) = \frac{4}{5} \cdot \frac{3}{8} = \frac{12}{40} = \frac{3}{10} = 0.3 \). Given \( P(A \cap B) = \frac{7}{40} = 0.175 \). Since \( 0.3
eq 0.175 \), events are not independent.

Step5: Part D

Calculate \( P(A) \cdot P(B) = \frac{7}{9} \cdot \frac{3}{4} = \frac{21}{36} = \frac{7}{12} \). Given \( P(A \cap B) = \frac{7}{12} \). Since \( \frac{7}{12} = \frac{7}{12} \), events are independent.

Answer:

A) Not Independent
B) Independent
C) Not Independent
D) Independent