# from necklace of three colors , probability that necklace does not have red bead

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 !

• 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).
– john
Dec 22, 2019 at 12:17
• 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 .
– John
Dec 22, 2019 at 13:08

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 $$\frac{79}{12483}$$