Question

11.3.73 Q20 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 five 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? The probability of at least one pair of neighbors using the same settings is approximately _____

Answer 1

In the transmitter there are 4 switches. Each switch can be on or off. So there are 2 choices each switch can be in.

Hence, the number of possible unique arrangements of these 4 switches (and hence the transmitter) = 2*2*2*2 = 8

Now suppose the 5 neighbors are A, B, C, D and E.

Lets calculate the number of ways in which these neighbors can receive the 8 transmitter arrangements.

Neighbor A can receive the 1 of the 8 possible arrangements. Since this arrangement can be repeated for neighbor B, we are still left with 8 arrangements now for neighbor B.

Neighbor B can receive the 1 of the 8 arrangements. Since this arrangement can be repeated for neighbor C, we are still left with 8 arrangements now for neighbor C.

Neighbor C can receive the 1 of the 8 arrangements. Since this arrangement can be repeated for neighbor D, we are still left with 8 arrangements now for neighbor D.

Neighbor D can receive the 1 of the 8 arrangements. Since this arrangement can be repeated for neighbor E, we are still left with 8 arrangements now for neighbor E.

Neighbor E can receive 8 of the arrangements.

Hence, the total number of ways in which these 5 neighbors can have these 8 arrangements = 8*8*8*8*8

Note that this is a case of permutation with repetition. Here we have to find the number of ways of choosing between n (8 here) things, taking r (5 here) at a time, where repetition is allowed. This is equal to n^r or 8^5.

Now we will calculate the number of ways in which each neighbor receives a different arrangement of transmitter switches.

Neighbor A can receive the 1 of the 8 possible arrangements. Since each neighbor has to have a different arrangement, we are left with 7 arrangements now for neighbor B.

Neighbor B can receive the 1 of the 7 arrangements. Since each neighbor has to have a different arrangement, we are left with 6 arrangements now for neighbor C.

Neighbor C can receive the 1 of the 6 arrangements. Since each neighbor has to have a different arrangement, we are left with 5 arrangements now for neighbor D.

Neighbor D can receive the 1 of the 5 arrangements. Since each neighbor has to have a different arrangement, we are left with 4 arrangements now for neighbor E.

Neighbor E can receive 4 of the arrangements.

Hence, the number of ways in which these 5 neighbors can receive these 8 arrangements, such that the arrangement of each neighbor is different = 8*7*6*5*4

Note that this is a case of permutation without repetition. Here we have to find the number of ways of choosing between n (8 here) things, taking r (5 here) at a time, where repetition is not allowed. This is equal to

Now we will calculate the number of ways in which atleast 1 or more neighbor have the same arrangement of switches.

Note that, the number of ways that these 5 neighbors receive these arrangements, such that atleast 1 pair of neighbors receives the same arrangement

= The total number of ways in which these 5 neighbors can have these 8 arrangements - The number of ways in which these 5 neighbors can receive these 8 arrangements, such that the arrangement of each neighbor is different

= 8*8*8*8*8 - 8*7*6*5*4

Probability(atleast 1 pair of neighbors receives the same arrangement) = (Number of ways that these 5 neighbors receive these arrangements, such that atleast 1 pair of neighbors receives the same arrangement) /  (Total number of ways in which these 5 neighbors can have these 8 arrangements)

= (8*8*8*8*8 - 8*7*6*5*4)/(8*8*8*8*8) = 0.7949.