# Thread: group theory

1. ## group theory

Hi guys, just a question from an old crpytography exam im having troubles with, thanks for the help=)

Let G be a finite cyclic group and let g be a generator for G. Suppose |G|=12.

Six elements of G are chosen at random (with replacement). (i)Find the probability that at least one is a generator.
(ii)Show that the probability that two or more are the same is at least 0.77

(i) P(at least 1 generator) = 1 - P(not getting a generator)
= 1-(2/3)^6
= 0.912

(ii) Really need help with this thanks

2. Originally Posted by james12
Hi guys, just a question from an old crpytography exam im having troubles with, thanks for the help=)

Let G be a finite cyclic group and let g be a generator for G. Suppose |G|=12.

Six elements of G are chosen at random (with replacement). (i)Find the probability that at least one is a generator.
(ii)Show that the probability that two or more are the same is at least 0.77

(i) P(at least 1 generator) = 1 - P(not getting a generator)
= 1-(2/3)^6
= 0.912

(ii) Really need help with this thanks

Well, what's the probability that all 6 elements chosen that way are different? It is

$\displaystyle \frac{11}{12}\cdot\frac{10}{12}\cdot\frac{9}{12}\c dot\frac{8}{12}\cdot\frac{7}{12}=0.2228009...$ , so...

Tonio