Let ω be a complex cube root of unity with ω ≠ 1. A fair die is thrown three times. If r1, r2 and r3 are the numbers obtained on the die then the probability that is

Engineering

Mathematics

Conditional Probability

Question

Let ω be a complex cube root of unity with ω ≠ 1. A fair die is thrown three times. If r₁, r₂and r₃ are the numbers obtained on the die then the probability that $ω^{r_{1}} + ω^{r_{2}} + ω^{r_{3}} = 0$ is

$\frac{1}{18}$

$\frac{1}{9}$

$\frac{2}{9}$

$\frac{1}{36}$

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$r 1, r 2, r 3 \in {1, 2, 3, 4, 5, 6}$
$r 1, r 2, r 3$ are if the form 3k 3k + 1, 3k + 2
Required probability $\frac{3! \times^{2} C_{1} \times^{2} C_{1} \times^{2} C_{1}}{6 \times 6 \times 6} = \frac{6 \times 8}{216} = \frac{2}{9}$

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