91Ó°ÊÓ

Combinations are a key concept in probability, especially when the order of selection doesn't matter. In our problem with modems, the task is to select 4 out of 20 modems.
The formula for combinations is: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] Here, \ {n\} is the total number of items (20 modems) and \ {k\} is the number of items to choose (4 modems).
Using this formula gives us the total number of ways to choose 4 modems from 20. This understanding is crucial because it forms the base of calculating probabilities in our scenario.

In probability exercises, defective items represent the unfavored outcomes we want to avoid. For our shipment of 20 modems, there are 3 defective ones.
When calculating probabilities, we need to know both the number of defective items and the total items. This helps us determine the favorable conditions. In our modem case, avoiding these 3 defective items is essential to determining if the shipment is accepted.
This concept is important because it directly influences the number of favorable outcomes available, impacting the probability calculations.

Calculating probability involves determining the ratio of favorable outcomes to the total number of possible outcomes.
For our modem problem, the probability of selecting 4 working modems is given by: \[ P(\text{all 4 work}) = \frac{C(17, 4)}{C(20, 4)} \] Here, \ {C(17, 4)\} represents the number of ways to choose 4 working modems from the 17 non-defective ones.
Likewise, \ {C(20, 4) \} is the total number of ways to choose any 4 modems from the 20 available.
Dividing these values simplifies the complex task of finding probability down to a manageable calculation. This method ensures accuracy and clarity in our solution.

Non-defective items are those in the group that meet the quality standards. In this exercise, there are 17 non-defective modems out of the total 20.
Calculating the favorable outcomes requires us to focus on the non-defective items. For selecting 4 working modems, we use the combinations formula to find the number of ways to pick from these 17 good modems: \[ C(17, 4) = \frac{17!}{4!(17-4)!} \]
This gives us the total possible ways to pick 4 non-defective ones. This number is critical for comparing against the total combinations, thereby forming the basis of our probability calculations.
Focusing on non-defective items ensures that only the acceptable configurations are counted in our favorable outcomes.