How to add sounds to the logic probe

Resistors in volume in China cost about 0.1 cent US.
Electrolytic capacitors cost around 1 cent up to around 33 uF.
Some selected ICs cost as little as 2 cents 0 eg TL431, a dual opamp such as LM358 may be 5 cents.

An NE567 would do your job but is a bit rare and may cost 10 - 20 cents.
NJM567 datasheet
I had altogether too much to do with these in another lifetime. They work quite well.

• The NJM567 tone and frequency decoder is a highly stable phase locked loop with synchronous AM lock detection and power output circuitry. Its primary function is to drive a load whenever a sustained frequency within its detection band is present at the self-biased input. The bandwidth cebter frequency, and output delay are independently determined by means of four external components.

A CD4046 digital PLL would do your job - pricing here and datasheet here - TI and here -Fairchild

Price from Digikey is $US12.8 at 50,000 quantity so Asian price may be 5 cents.

Circuit diagram in real world use is not much more than shown below. Phase comparator two is just an XOR gate but works like magic. The circuit shown below would probably cost under 10 cents in 10,000 + volume in China but PCB cost and manufacturing costs will about double that.

This circuit will work superbly with minimal noise and quite well in modestly noisy circumstances. In very noisy environments a 5th order active filter can be added at a cost of 2 transistors and RC glue - but good quality R's and 's needed for a stable filter - say an additional $0.10.

A simple circuit to do this would use a single transistor, one resistor and a buzzer. Connect two 1.5 volt batteries in series to get 3 volts. Connect one end of a 10 kilohm resistor to the positive terminal of the batteries and the other end to the base of a general purpose NPN transistor (2N2222, 2N3904, etc.). Connect the negative end of the battery to the emitter of the transistor. Connect one wire of the buzzer to the positive end of the battery and the other wire to the collector of the transistor. If the buzzer has polarity markings, follow them: positive to positive end of battery, negative to transistor collector. Connect your sensing wire from the base to the emitter of the transistor. As long as the wire is connected , it will short the base to the emitter and prevent the transistor from turning on. When it is cut, the batteries will send current to the base of the transistor through the resistor. This will turn on the transistor which means the collector to emitter voltage will be very small and most of the battery voltage will be across the buzzer which will then turn on. With the sense wire connected, the batteries only have to provide current through the resistor which will be about 3 volts divided by 10 kilohms or 0.3 ma. Two AA batteries can provide this much current for hundreds of hours. If necessary you can use C or D batteries for even greater lifetime. This circuit is simply and can be easily modified to handle other sound sources if needed.