Probability
Complex Numbers and Probability
Grade 12
Question:
<p>Let \(\omega\) be a complex cube root of unity with \(\omega \neq 1\). A fair die is thrown three times. If \(r_1, r_2\) and \(r_3\) are the numbers obtained on the die, then the probability that \(\omega^{r_1} + \omega^{r_2} + \omega^{r_3} = 0\) is</p>
<p>(a) \(\frac{1}{18}\)</p>
<p>(b) \(\frac{1}{9}\)</p>
<p>(c) \(\frac{2}{9}\)</p>
<p>(d) \(\frac{1}{36}\)</p>
Step-by-Step Solution
Key Concept: Use properties of cube roots of unity: $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$. Classify die outcomes by residue modulo 3.
<p>Not provided in source text</p>
Correct Answer: b