Let omega be a complex cube root of unity with omega ≠ 1 . A fair die is thrown three times. If r?

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14.Probability

normal

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 $\omega^{I_1}+\omega^{\mathrm{I}_2}+\omega^{\mathrm{I}_3}=0$ is

$\frac{1}{18}$

$\frac{1}{9}$

$\frac{2}{9}$

$\frac{1}{36}$

(IIT-2010)

Solution

$ \mathrm{r}_1, \mathrm{r}_2, \mathrm{r}_3 \in\{1,2,3,4,5,6\} $

$ \mathrm{r}_1, \mathrm{r}_2, \mathrm{r}_3 \text { are of the form } 3 \mathrm{k}, 3 \mathrm{k}+1,3 \mathrm{k}+2 $

$ \text { Required probability }=\frac{3!\times{ }^2 \mathrm{C}_1 \times{ }^2 \mathrm{C}_1 \times{ }^2 \mathrm{C}_1}{6 \times 6 \times 6}=\frac{6 \times 8}{216}=\frac{2}{9}$

Standard 11

Mathematics

Let a computer program generate only the digits $0$ and $1$ to form a string of binary numbers with probability of occurrence of $0$ at even places be $\frac{1}{2}$ and probability of occurrence of $0$ at the odd place be $\frac{1}{3}$. Then the probability that $'10'$ is followed by $'01'$ is equal to :

hard

(JEE MAIN-2021)

Three randomly chosen nonnegative integers $x, y$ and $z$ are found to satisfy the equation $x+y+z=10$. Then the probability that $z$ is even, is

normal

(IIT-2017)

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medium

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medium

(JEE MAIN-2021)

Let a die be rolled $n$ times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is $\frac{ k }{2^{15}}$, then $k$ is equal to: