Sudheer has 3 identical white cubes and wants to turn them into dice. For each cube he makes two independent decisions:
1. Colour: Red or Blue (each with probability $\frac{1}{2}$).
2. Pattern: Dots or Digits (each with probability $\frac{1}{2}$).
He places all 3 dice in a bag. Renuka randomly selects one die and notes its colour and pattern. Define:
- Event $A$: The selected die is Red.
- Event $B$: The selected die is Dotted.
The probability that events $A$ and $B$ are independent is:
Step-by-Step Solution
Key Concept: Let $(x, y, z, w)$ be the number of dice of types (Rd, RD, Bd, BD). $A$ and $B$ are independent iff $P(A\cap B) = P(A)P(B)$, i.e., $\frac{x}{3} = \frac{x+y}{3}\cdot\frac{x+z}{3}$, or $3x = (x+y)(x+z)$. Enumerate over all outcomes $(x,y,z,w)$ with $x+y+z+w=3$.
Enumerate solutions to $3x=(x+y)(x+z)$ with $x+y+z+w=3$, $x,y,z,w \geq 0$. Favourable cases: $x=0$ (then $0=0$, always true); and specific positive $x$ cases. Total favourable = 28 out of 64. Probability $= \frac{28}{64} = \frac{7}{16}$.
Correct Answer: A