A box contains V balls of which N[itex]_{g}[/itex] are green, N[itex]_{r}[/itex] are red and V-N[itex]_{g}[/itex]-N[itex]_{r}[/itex]=N[itex]_{b}[/itex] are blue.(adsbygoogle = window.adsbygoogle || []).push({});

Now somebody shakes the box vigorously, brings it to rest and then observes the

arrangement of the balls in the box. Suppose this is repeated many times so that

probabilities of different ball configurations can be defined as frequencies of occurrence.

What is the probability, P(G, R) that there are G green balls and R red balls in the

left half of the box? (Assume G+R<V/2)

I believe this probability is equal to [itex]\Gamma[/itex](G, R)/[itex]\Gamma[/itex](total)...

In this case, I think [itex]\Gamma[/itex](G, R) would be:

[itex]\Gamma[/itex](G, R) = (([itex]\frac{(N_g)!}{(N_g)!(V-N_r-N_b)!}[/itex][itex])^2)((\frac{(N_r)!}{(N_r)!(V-N_g-N_b)!}[/itex])^2)(([itex]\frac{V}{2}[/itex]!)^2)

And [itex]\Gamma[/itex](total) is V!.

Does this look right? In the first expression, the first term represents the number of ways to select which green balls are on the left hand side times the number of ways to select which red balls are on the left hand side, the second term represents the number of ways to select which red balls are on the left hand side times the number of ways to select which green balls are on the left hand side, and the third term represents the number of ways to arrange V/2 balls on the LHS times the number of ways to arrange V/2 balls on the RHS.

Am I on the right track?

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# Balls in a box shaking experiment

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