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Imagine a node "I" with two children, "W" and "H". "I" means that roads are icy, and "W" means that Watson crashes. "H" means Holmes crashes.

If I wanted to know the probability of the roads being icy and Watson crashing, which formula should I be using? I would be looking for P(I and W).

For dependent events, P(A n B) = P(A) * P(B|A).

For independent events, P(A n B) = P(A) * P(B).

So my confusion is with the fact that it's a Bayesian Network. How does the dependency and Independence come into play? Thanks!

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The probability of the conditions being icy and Watson crashing is equal to the prior probability that it is icy times the probability that if it is icy Watson will crash:

$P(I \land W) = P(W|I)P(I)$.

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So my confusion is with the fact that it's a Bayesian Network. How does the dependency and Independence come into play? Thanks!

In a Bayesian network, the branches indicate dependence of the nodes. So children are dependent on their parents but (conditionally) independent of each other; unless there's some incestuous branching in the network.

So the diagram means that the events $W$ and $H$ are conditionally independent when given $I$, but each is dependent on $I$. $$\require{enclose}\enclose{circle}{\,\rm W\,}\raise{2.75ex}{\swarrow\raise{2.5ex}{\enclose{circle}{\;\rm I\;}}\searrow}\enclose{circle}{\,\rm H\,}$$

Thus we have the factorisation: $$\mathsf P(I\cap W)=\mathsf P(I)\,\mathsf P(W\mid I)$$

The diagram will usually be accompanied with tables that provide values for these factors.

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