Question 1164269
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            Wording seems to be complicated,  but the problem is,  actually,  quite simple.



<pre>
With 4 switches, each of which can be in any of the two possible positions,

there are  {{{2^4}}} = 16  different setting for the control.



So, the probability for every one single of 4 neighbors to have the same setting in his (or her) control is  {{{1/16}}}
(their controls are of the same construction (!) ).



The probability that NO ONE of 4 neighbors has the same setting is  {{{(15/16)^4}}}.


The probability under the question is the COMPLEMENT to it, i.e.


    P = 1 - {{{(15/16)^4}}} = 1 - 0.7225 = 0.2275.     <U>ANSWER</U>
</pre>

Solved.


Very nice probability problem &nbsp;(!)


Hope that I correctly use the technical terms in my post . . . 



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