Question

a red 8 - sided die and a green 8 - sided die are rolled. let a be the event that the red die comes up odd, and let b be the event that the green die comes up odd. are these events dependent or independent?
find ( p(a) )
( p(a)=\frac{1}{2} ) (simplify your answer.)
find ( p(a mid b) )
( p(a mid b)=square ) (simplify your answer)

Explanation:

Step1: Understand the independence of events

If two events \(A\) and \(B\) are independent, then \(P(A|B)=P(A)\).

Step2: Analyze the nature of die - rolling events

When rolling a red 8 - sided die and a green 8 - sided die, the outcome of the green die (event \(B\)) has no impact on the outcome of the red die (event \(A\)).

Step3: Calculate \(P(A|B)\)

Since the events are independent, \(P(A|B)=P(A)\)
We know that for an 8 - sided die, the odd numbers are \(1,3,5,7\). So \(n(\text{odd}) = 4\) and \(n(\text{total})=8\).
\(P(A)=\frac{4}{8}=\frac{1}{2}\)

Answer:

\(P(A|B)=\frac{1}{2}\)