Knotty Marble Probability

When Aditya takes at the end, the other two MUST have $2$ same and $3$ different marbles. And Aditya must pick one of the $2$ same marble pair from both of them with $\dfrac{2}{5}\times\dfrac{2}{5}=\dfrac{4}{25}$ chance.

This is common and must be multiplied at the end to all possible scenarios.

$(A)$ Satvik picks same colour from both with $\dfrac{1}{4}$ chance. Now he has $3$ same and $3$ different marbles.

Agnishom must pick one of the $3$ same with $\dfrac{3}{6}$ chance and can pick any colour from Aditya.

So its $\dfrac{1}{4}\times\dfrac{3}{6}=\dfrac{1}{8}$ chance.

$(B)$ Satvik picks different coloured marbles from both with $\dfrac{3}{4}$ chance. Now he has $2$ same, $2$ same and $2$ different marbles.

Agnishom must pick from either of the $2$ sames.

$(1)$ If Agnishom picks the colour he needs to make it $4$ differently coloured marbles from Satvik with $\dfrac{2}{6}$ chance, then he can pick any colour from Aditya.

So its $\dfrac{3}{4}\times\dfrac{2}{6}=\dfrac{1}{4}$ chance.

$(2)$ If Agnishom picks a colour from Satvik that he already has with a $\dfrac{2}{6}$ chance, then he must pick the colour he lacks from Aditya with $\dfrac{1}{3}$ chance.

So its $\dfrac{3}{4}\times\dfrac{2}{6}\times\dfrac{1}{3}=\dfrac{1}{12}$ chance.

Adding $\dfrac{1}{8}+\dfrac{1}{4}+\dfrac{1}{12}=\dfrac{11}{24}$

Multiplying this by Aditya's chance we get probability of the event as

$\dfrac{11}{24}\times\dfrac{4}{25}=\dfrac{11}{150}$