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I know that Kirchoff's law comes from the fact that charge is conserved at every junction of a circuit. But is it useful (and true) to think about this in terms of probabilities? As in, the probabilities that a current splits at a junction and goes in one direction or the other add up to 1. For example, suppose that a current $ I_1 $ splits into $ I_2 $ and $ I_3 $ going in opposite directions. Then $ \frac{I_2}{I_1} + \frac{I_3}{I_1} = 1 $.

Apologies if I sound rather ignorant. I'm just curious.

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No, Kirchoff's first law is based on the conservation of charge and the assumption that the charge in a circuit doesn't build up anywhere and stays evenly distributed.

In some small amount of time a total amount of charge comes into the junction, $Q_{in} = I_1\Delta t$. In order for the charge not to pile up at the junction, the same amount of charge has to leave the junction in the same amount of time. $Q_{out} = I_2\Delta t + I_3\Delta t$. Since the charge in has to equal the charge out $I_1\Delta t = (I_2 + I_3)\Delta t$ or $I_1 = I_2 +I_3$.

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