Question:

Step 1: To find the value of $p$ when $A$ and $B$ are mutually exclusive, we use the formula for the probability of the union of two mutually exclusive events: $P(A \cup B) = P(A) + P(B)$. Given that $P(A) = 0.5$ and $P(A \cup B) = 0.8$, we can solve for $P(B)$, which in this context is equal to $p$. So, we have $0.8 = 0.5 + p$. $$ 0.8 = 0.5 + p $$ Solving for $p$, we get: $$ p = 0.8 - 0.5 = 0.3 $$ Step 2: Next, to find the value of $q$ when $A$ and $B$ are independent, we use the formula for the probability of the union of two independent events: $P(A \cup B) = P(A) + P(B) - P(A)P(B)$. Substituting the given values, we have $0.8 = 0.5 + q - 0.5q$. This equation will allow us to solve for $q$. $$ 0.8 = 0.5 + q - 0.5q $$ Step 3: We simplify the equation from Step 2 to solve for $q$. First, we combine like terms: $$ 0.8 = 0.5 + 0.5q $$ Then, we isolate the terms involving $q$ on one side: $$ 0.8 - 0.5 = 0.5q $$ $$ 0.3 = 0.5q $$ Solving for $q$, we get: $$ q = \frac{0.3}{0.5} = 0.6 $$ Step 4: Finally, to find $q/p$, we divide the value of $q$ by the value of $p$ obtained from the previous steps. $$ \frac{q}{p} = \frac{0.6}{0.3} = 2 $$ Therefore, the final answer is $2$.