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 six 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?

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 six neighbors with the same type of opener set their switches independently. The probability of at least one pair of neighbors using the same settings is 0.65633

Step-by-step explanation:

Step 1

In the question it is given that

Automatic garage door opener utilizes a transmitter control with four independent switches

So. the number of Combinations possible with the Transmitters =

2*2*2*2 = 16

Step 2

Probability of at least one pair of neighbors using the same settings = 1 - Probability of All Neighbors using different settings.

= 1 - 16*15*14*13*12*11 / (16^6)

Step 3

Probability of at least one pair of neighbors using the same settings=

= 1 - 0.343666

Step 4

So the probability of at least one pair of neighbors using the same settings

is 0.65633