probability that the light is on, 5 switches, circuit diagram
Source: Netherlands - Dutch MO 1984 p2
December 25, 2022
combinatoricsprobability
Problem Statement
The circuit diagram drawn (see figure ) contains a battery , a lamp and five switches to . The probability that switch is closed (makes contact) is , for the other four switches that probability is (the probabilities are mutually independent). Calculate the probability that the light is on.[asy]
unitsize (2 cm);draw((-1,1)--(-0.5,1));
draw((-0.25,1)--(1,1)--(1,0.25));
draw((1,-0.25)--(1,-1)--(0.05,-1));
draw((-0.05,-1)--(-1,-1)--(-1,0.25));
draw((-1,0.5)--(-1,1));
draw((-1,1)--(-0.5,0.5));
draw((-0.25,0.25)--(0,0));
draw((-1,0)--(-0.75,0));
draw((-0.5,0)--(0,0));
draw((0,1)--(0,0.75));
draw((0,0.5)--(0,0));
draw((-0.25,1)--(-0.5,1.25));
draw((-1,0.25)--(-1.25,0.5));
draw((-0.5,0.5)--(-0.25,0.5));
draw((0,0.75)--(0.25,0.5));
draw((-0.75,0)--(-0.5,-0.25));
draw(Circle((1,0),0.25));
draw(((1,0) + 0.25*dir(45))--((1,0) + 0.25*dir(225)));
draw(((1,0) + 0.25*dir(135))--((1,0) + 0.25*dir(315)));
draw((0.05,-0.9)--(0.05,-1.1));
draw((-0.05,-0.8)--(-0.05,-1.2));label("", (1.25,0), E);
label("", (-0.1,-1.1), SW);
label("", (-0.5,1.25), NE);
label("", (-1.25,0.5), SW);
label("", (-0.5,0.5), SW);
label("", (0.25,0.5), NE);
label("", (-0.5,-0.25), SW);
[/asy]