I need to find the probability of C and G. P(C and G). How can I calculate it? I got confused about how to deal with R.
2026-03-25 16:06:32.1774454792
Bayesian Network (Sprinkler Example), Rain and WetGrass prediction
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You deal with it through DAG Factorisation and the Law of Total Probability.
The directed acyclic graph's factorisation is: $\mathsf P_{N,R,C,G} =\mathsf P_N~\mathsf P_{R\mid N}~\mathsf P_{C\mid N}~\mathsf P_{G\mid R,C}$
Then you seek: $$\begin{align}\mathsf P(C{=}\mathrm v,G{=}\mathrm v)&=\sum_{n\in\{\mathrm{v,f}\}}\sum_{r\in\{\mathrm{v,f}\}}\mathsf P(N{=}n,R{=}r,C{=}\mathrm v,G{=}\mathrm v)\\[2ex]&=\sum_{n\in\{\mathrm{v,f}\}}\mathsf P(N{=}n)\mathsf P(C{=}\mathrm v\mid N{=}n)\sum_{r\in\{\mathrm{v,f}\}}\mathsf P(R{=}r\mid N{=}n)\mathsf P(G{=}\mathrm v\mid C{=}\mathrm v,R{=}r)\\&~~\vdots\end{align}$$