First slide

Introduction to probability

Question

Let $ω$ be a complex cube root of unity with $ω \neq 1$ . A fair die is thrown three times. If r₁, r₂ and r₃ are the numbers obtained on the die, then the probability that $ω^{r_{1}} + ω^{r_{2}} + ω^{r_{3}} = 0$ is

Moderate

A

1/18
B

1/9
C

2/9
D

1/36

Solution

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

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