Each bulb is modeled as an ohmic resistor of resistance R. In series, every bulb shares the one and only current path, so the same current I flows through all of them: I = Vs / (N·R), and each bulb's share of the source voltage is Vs/N — Kirchhoff's voltage law divides the loop's total EMF across N equal drops. Break or burn out any single bulb and the loop opens: I = 0 everywhere, and every bulb goes dark, not just the burnt one.
In parallel, each bulb is its own branch directly across the source, so by Kirchhoff's voltage law every active bulb sees the full Vs, independent of how many other bulbs are present. Burning one out just removes its branch — Kirchhoff's current law still balances for the rest, whose currents (I = Vs/R each) and brightness are completely unaffected. Total current is the sum of the active branches' currents, so total resistance actually drops as more bulbs are added in parallel.
Series: R_tot = N·R I = Vs/R_tot V_bulb = Vs/N P_bulb = I²R
Parallel: R_tot = R/N_active I_bulb = Vs/R V_bulb = Vs P_bulb = Vs²/R
Simplifications. Each bulb is a constant ohmic resistor. A real tungsten filament is strongly non-linear: its hot resistance is roughly ten times its cold resistance, which is why an incandescent bulb draws a surge when switched on. The conclusions about series and parallel wiring (shared current vs shared voltage, one open bulb breaking a series loop) do not depend on that. The source is ideal (no internal resistance) and wires have zero resistance. Independent check: the panel above re-solves the same circuit as a resistor network by Gaussian elimination (nodal analysis) and compares it with the closed-form formulas. Brightness of each bulb is rendered from its own computed power P (via emissive intensity and point-light output), so a dimmer bulb in series with a large N visibly glows weaker than a bright single parallel bulb at the same source voltage — the real, measurable trade-off between the two wiring schemes.