A planned community has 150 homes, each with an automatic garage door opener. The door opener has seven switches that a homeowner can set to 0 or 1. For example, a door opener code might be 1001100. Assuming all the homes in the community are sold, what is the probability that at least two homeowners will set their switches to the same code, and will therefore be able to open each other's garage doors?

Answer:The probability that at least two homeowners will set their switches to the same code is 100%.Step-by-step explanation:Consider the provided information.The total number of code can be set with 0 or 1 is:2×2×2×2×2×2×2=128There are 128 different codes.  The probability that code is unique is 1/128Now, the probability that all codes are unique is:[tex][\frac{1}{128}]^{150}[/tex]Hence, the probability that all the codes are not unique is:[tex]1-[\frac{1}{128}]^{150} \approx 1[/tex]Because the value of [tex][\frac{1}{128}]^{150}[/tex] is very smallAs they can set 128 different codes and there are 150 homes. So, at least two homeowners will set their switches to the same code is 100%.Hence, the probability that at least two homeowners will set their switches to the same code is 100%.