SOLUTION: Kevin installed a certain brand of automatic garage door opener that utilizes a transmitter control with four independent​ switches, each one set on or off. The receiver​ (wire

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Question 1164269: Kevin installed a certain brand of automatic garage door opener that utilizes a transmitter control with four independent​ switches, each one set on or off. The receiver​ (wired to the​ door) must be set with the same pattern as the transmitter.
If four neighbors with the same type of opener set their switches​ independently, what is the probability of at least one pair of neighbors using the same​ settings?
Answer by ikleyn(52887)   (Show Source): You can put this solution on YOUR website!
.

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


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

there are   = 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  
(their controls are of the same construction (!) ).



The probability that NO ONE of 4 neighbors has the same setting is  .


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


    P = 1 -  = 1 - 0.7225 = 0.2275.     ANSWER

Solved.

Very nice probability problem  (!)

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


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Please do not forget to post your "THANKS" for my teaching.



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