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Combinatorial Analysis is a crucial tool in probability and statistics. It helps us determine the number of possible outcomes in various scenarios. In our exercise with semiconductor chips, we need to find out how many ways we can choose a set number of chips from a lot.

Combinatorial analysis uses combinations since the order of selection does not matter. For example, when selecting two chips from 100, the number of combinations can be calculated using the formula:

• Combinations for selecting two chips from 100: \( \binom{100}{2} = \frac{100 \times 99}{2} \)
• This computes to a total of 4,950 possible combinations.

Similarly, when selecting three chips:

• Combinations for selecting three chips: \( \binom{100}{3} = \frac{100 \times 99 \times 98}{6} \)
• Leading to 161,700 possible outcomes.

By finding these combinations, we establish the groundwork to calculate probabilities accurately. This is essential in solving problems where understanding all potential outcomes is needed to find precise probabilities.

Defective probability refers to the likelihood of selecting an item that is defective from a group. When more than one item is selected, such as with our semiconductor chips, it involves considering both the defective items and the method of selection without replacement.

In the exercise, for part (a), we calculate the likelihood that the second chip selected is defective. This involves two situations:

• The first chip is non-defective; the second is defective, giving us \( 80 \times 20 = 1600 \) favorable outcomes.
• Both chips are defective, calculated with \( \binom{20}{2} = 190 \) favorable outcomes.

Adding these gives us a total of 1,790 favorable combinations for part (a).

For part (b), where we want all three chips to be defective, we look at selecting 3 defective ones out of 20.

• This results in \( \binom{20}{3} = 1140 \) favorable outcomes.

Understanding these scenarios helps us see how probabilities can be structured around the defectiveness of items, informing quality control and risk assessment decisions.

Statistical Probability is the foundation of evaluating uncertain events. It is calculated as the ratio of favorable outcomes to the total outcomes. In the semiconductor chip example, calculating statistical probability allows us to determine the likelihood of selecting defective chips from the lot.

For part (a), to find the probability that the second chip is defective, we use the formula:

• \( P(\text{second is defective}) = \frac{1790}{4950} = \frac{179}{495} \approx 0.3616 \)

This probability outcome means there is about a 36.16% chance that the second selected chip is defective.

Meanwhile, part (b) involves the probability that all three selected chips are defective:

• \( P(\text{all defective}) = \frac{1140}{161700} \approx 0.00705 \)

This result shows a mere 0.705% chance of all three chips being defective, indicating a very low likelihood. Statistical probability therefore provides a quantitative measure of uncertainty, helping in making informed decisions based on empirical data.