Question

An inspector is doing a quality control assessment on a new shipment of wireless headphones. The original invoices for the headphones show 35% of their supply is from the Jabra company and 65% is from the Samsung company. The inspector also knows that Jabra’s headphones have a 3% defect rate (meaning that 3% of the sample will fail on first usage). The defect rate for Samsung headphones is estimated at 4%.

Complete each subquestion that follows, showing all of your work either in this screen OR compiled into a separate word/excel document and submitted to the Scratch work dropbox. Further, clearly note each subquestion you answer.

1. Define the two “events” of interest here. (2 points)
2. List all of the information that is explicitly stated in the prompt using correct probability notation. (2 points)
3. List all of the information that is not stated in the prompt, but that you can infer from existing information using correct probability notation. (2 points)
4. Draw a probability tree diagram that reflects this situation. (4 points) - note that you can add in images directly to the response window using the image icon above.
5. What is the probability of a randomly selected pair of headphones from this supply being Jabra brand and failing on first use? Show all of your work to receive credit. (2 points)
6. What is the probability of a randomly selected pair of headphones from this supply not failing upon first use? Show all of your work to receive credit. (3 points)

Answer 1

We are given here that: 35% of their supply is from the Jabra company and 65% is from the Samsung company. Therefore,
P(J) = 0.35, and P(S) = 0.65

The inspector also knows that Jabra’s headphones have a 3% defect rate (meaning that 3% of the sample will fail on first usage). The defect rate for Samsung headphones is estimated at 4%. Therefore, we have here:
P(D | J) = 0.03,
P(D | S) = 0.04

The two events of interest here are:
a) From which company is the supply from - Jabra or Samsung
b) Whether the headphone is defective or not.

The probability notations are already described above as:
P(J) = 0.35, and P(S) = 0.65
P(D | J) = 0.03,
P(D | S) = 0.04

The other probabilities that could be derived here are given as:

P(ND | J) = 1 - P(D | J) = 1 - 0.03 = 0.97, which is the probability of a non defective headphone given a Jabra headphone is selected.
P(ND | S) = 1 - P(D | S) = 1 - 0.04 = 0.96, which is the probability of a non defective headphone given a Samsung headphone is selected.

The Tree Diagram here is given as:

Now the probability of a randomly selected pair of headphones from this supply being Jabra brand and failing on first use is computed using Bayes theorem here as:

P(J and D) = P(D | J)P(J) = 0.03*0.35 = 0.0105

Therefore 0.0105 is the required probability here.

Now using addition law of probability, we have here:
P(D) = P(D | J)P(J) + P(D | S)P(S) = 0.0105 + 0.65*0.04 = 0.0365

Therefore, P(ND) = 1 - P(D) = 1 - 0.0365 = 0.9635

Therefore the probability of a randomly selected pair of headphones from this supply not failing upon first use is given as 0.9635