2. Bender electronics buys keyboards for its computers from another company. The keyboards are received in shipments of 100 boxes, each box containing 20 keyboards. The quality control department at Bender Electronics first randomly selects one box from each shipment and then randomly selects 5 keyboards from that box. The shipment is accepted if not more than 1 of the 5 keyboards is defective. The quality control inspector at Bender Electronics selected a box from a recently received shipment of keyboards. Unknown to the inspector, this box contains 6 defective keyboards.
a) What is the probability that this shipment will be accepted?
b) What is the probability that this shipment will not be accepted?
Bender Electronics buys keyboards for its computers from another company. The keyboards are received
in shipments of 100 boxes, each box containing 20 keyboards. The quality control department at
Bender Electronics first randomly selects one box from each shipment and then randomly selects 5 keyboards
from that box. The shipment is accepted if not more than 1 of the 5 keyboards is defective. The
quality control inspector at Bender Electronics selected a box from a recently received shipment of keyboards.
Unknown to the inspector, this box contains 6 defective keyboards.
a. What is the probability that this shipment will be accepted?
b. What is the probability that this shipment will not be accepted?
To answer these questions, we need to calculate the probabilities using the information provided.
a) To find the probability that the shipment will be accepted, we need to determine the probability of selecting at most 1 defective keyboard out of 5.
Given that there are 6 defective keyboards in the box, the probability of selecting a defective keyboard is 6/20 (since there are 20 keyboards in each box). The probability of selecting a non-defective (good) keyboard is 1 - (6/20) = 14/20.
We are selecting 5 keyboards from the box. The probability of selecting 0 defective keyboards can be calculated using the binomial distribution formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- n = number of trials (5 in our case)
- k = number of successes (0 in this case, as we want to select 0 defective keyboards)
- p = probability of success (probability of selecting a good keyboard = 14/20)
P(X = 0) = C(5, 0) * (14/20)^0 * (6/20)^(5 - 0) = (6/20)^5 ≈ 0.07776
Similarly, we can calculate the probability of selecting exactly 1 defective keyboard:
P(X = 1) = C(5, 1) * (14/20)^1 * (6/20)^(5 - 1) = 5 * (14/20)^1 * (6/20)^4 ≈ 0.2592
To calculate the probability of accepting the shipment, we add the probabilities of selecting 0 and 1 defective keyboards:
P(accepted) = P(X = 0) + P(X = 1) ≈ 0.07776 + 0.2592 ≈ 0.33696
Therefore, the probability that this shipment will be accepted is approximately 0.33696.
b) To find the probability that the shipment will not be accepted, we need to calculate the probability of selecting more than 1 defective keyboard out of 5.
P(not accepted) = 1 - P(accepted) ≈ 1 - 0.33696 ≈ 0.66304
Therefore, the probability that this shipment will not be accepted is approximately 0.66304.