What is the probability of a mated triple with three sexes in sea-creatures?

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Homework Statement



A new lifeform has been discovered in the sea. In particular, these sea-creatures
come in THREE sexes, which are indistinguishable (except by other sea-creatures). If A robot captures N specimens, what is the probability these
chocies will allow a “mated triple” with its promise of more little sea-creatures to come?


Homework Equations


This homework is supposed to be on conditional probabilities and counting methods, so I suppose Baye's Theorem and Law of Total Probabilities would be useful


The Attempt at a Solution


Unfortunately, I truly have no idea where to begin this problem.
Any ideas?
 
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I suspect you are to assume the genders are equally likely in the sea-creature population.

I would first try this indirectly but trying to calculate the probability that all N creatures belong to two identical genders.

--Elucidus
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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