Consider the design of a triply redundant majority voting system, with three binary inputs...

Consider the design of a triply redundant majority voting system, with three binary inputs a, b, and c shown below. The inputs represent data from three independent sources from which the objective is to determine if the majority of the input bit values are logic level 0 or 1. Each majority circuit outputs the bit value that is in the majority of the inputs. The output of each majority circuit is fed into a resistor and LED (light-emitting diode) network. The LEDs are lit if the output of the majority circuit is a logic 1, otherwise the LED is off. Ideally, all three LEDs are lit if a majority of inputs are 1, else they are all off. However, if part of the system fails, the LED readings may not be reliable, so a majority-rules decision is used on the LEDs. The criterion used is that if two or more LEDs are on, then a majority of inputs are considered to be high, otherwise, a majority of inputs are considered to be low. Determine the probability of a false reading based upon this criterion if each component in the system (gate, resistor, and LED) has a reliability of 90%.