Circuit completes whether base current applied or not - transistor
Assuming battery + is on the transistor/R1 connection, and battery - to the lamp/Probe 2 connection:

By disconnecting the wire going to the 100K you are only removing the part of the circuit that tries to ensure the transistor stays off when it should. The 100K makes the circuit a bit less sensitive so that only a somewhat lower resistance short (between the probes) turns on the lamp (or LED). With out the 100K connection there is a chance that the lamp turns on slightly if you just short the probes with your hands.

But even with out that connection to the 100K, shorting probe 1 to probe 2 will still pull current through the emitter/base connection of the transistor, (with the path being from the battery, to emitter, to base, through R2, probe 1, probe2, and back to the battery.)  This turns on the emitter to collector path, turning on the lamp.