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Consider the collection of all possible necklaces that one can make with 13 beads where for each bead there are three possible colors Red, Yellow ,Green . If one picks a necklace at random what is the probability that it does not contain any red bead ?

Is there any method of solving this using basic counting principles ? Any insight is appreciated !

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  • $\begingroup$ Absolutely! And this particular question doesn’t rely on the beads forming a necklace - you’re just asking about the probability that a random collection of 13 beads doesn’t contain a red one. That’s $(2/3)^(13)$ since the probability of not picking red each time is $2/3$ (assuming equal probability for all colors). $\endgroup$
    – john
    Dec 22, 2019 at 12:17
  • $\begingroup$ I am not getting what do you want to say ? What you are saying is true if we are arranging beads in a straight line . But we are making a necklace out of it . $\endgroup$
    – John
    Dec 22, 2019 at 13:08

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This is quite a sophisticated problem and probably the best way to solve it is to use Burnside's Lemma to count the number of essentially different necklaces.

With $3$ colours there are $3^{13}$ arrangements which can be reflected (turned over) and rotated.

The identity transformation fixes all $3^{13}$ arrangements.

The other $12$ rotations each fix just $3$ necklaces (the ones which use just one colour).

The $13$ reflections each fix $3^7$ necklaces ('opposite' beads must have the same colour).

So the number of essentially different necklaces is $$\frac{3^{13}+12\times3+13\times3^7}{26}=62415.$$ With just $2$ colours the number of essentially different necklaces is $$\frac{2^{13}+12\times2+13\times2^7}{26}=380.$$

The probability is $\frac{79}{12483}$

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