Bender Electronics buys keyboards for its computers from another company. The keyboards are received in shipments of 75 boxes, each box containing 30 keyboards. The quality control department at Bender Electronics first randomly selects one box from each shipment and then randomly selects 8 keyboards from that box. The shipment is accepted if not more than 1 of the 8 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 9 defective keyboards.
a. What is the probability the shipment will be accepted?
b. What is the probability the shipment will not be accepted?
To calculate the probability that the shipment will be accepted, we need to determine the number of ways the quality control inspector can select 8 keyboards from the box, given that 9 of the keyboards are defective.
Step 1: Calculate the total number of ways to choose 8 keyboards from a box of 30 keyboards.
This can be calculated using the combination formula, which is represented as "nCr" and calculates the number of ways to choose "r" items from a set of "n" items. In this case, n = 30 (total number of keyboards in the box) and r = 8 (number of keyboards the inspector selects).
nCr = n! / (r! * (n-r)!)
Substituting the values:
30C8 = 30! / (8! * (30-8)!)
Step 2: Calculate the number of ways to choose 8 non-defective keyboards from the box.
Since there are 9 defective keyboards in the box, we subtract this value from the total number of ways calculated in Step 1.
Number of ways to choose 8 non-defective keyboards = 30C8 - 9C8
Step 3: Calculate the probability that the shipment will be accepted.
The shipment will be accepted if not more than 1 of the 8 selected keyboards is defective. In other words, the shipment will be accepted if all 8 selected keyboards are non-defective.
Probability of selecting 8 non-defective keyboards = Number of ways to choose 8 non-defective keyboards / Number of ways to choose 8 keyboards
a. Therefore, the probability that the shipment will be accepted is:
Probability of acceptance = (30C8 - 9C8) / 30C8
To calculate the probability that the shipment will not be accepted, we simply subtract the probability of acceptance from 1 (since there are only two possible outcomes: acceptance or non-acceptance).
b. Probability of non-acceptance = 1 - Probability of acceptance
Now, let's calculate these probabilities.
Note: Calculations involving factorials can be time-consuming, so I would recommend using a scientific calculator or an online calculator that can handle large numbers.