Question

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?

Answer 1

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

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