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In probability theory, a sample space is a fundamental concept that refers to the set of all possible outcomes of a particular experiment. In our injection molding example, each outcome corresponds to a specific cavity in the mold where a part can form. Since there are 8 cavities, our sample space is represented as the set:

• \( \{1, 2, 3, 4, 5, 6, 7, 8\} \)

Think of sample space as the universe of possibilities for a given scenario. Each element in this space signifies a potential result, and understanding the sample space helps in determining various probabilities associated with the experiment. For instance, it sets the foundation for identifying all potential sources of a part in the molding process.

Favorable outcomes are essentially the specific results we are interested in when calculating probability. For any probability question, favorable outcomes are those elements of the sample space that satisfy a certain condition.
For example, if we want to find the probability that a part is produced from cavity 1 or 2, these become our favorable outcomes:

• Cavity 1
• Cavity 2

Similarly, if we assess a condition where a part is not produced from cavity 3 or 4, then the favorable outcomes are those cavities not included in the condition, like 1, 2, 5, 6, 7, and 8.
By accurately identifying these outcomes, we can then calculate probabilities by dividing the number of favorable outcomes by the total number in the sample space.

Injection molding is a popular manufacturing process used to form parts by injecting material into a mold. The mold has multiple cavities which produce identical parts in one operation.
This process is significant in mass production because it allows the creation of many parts with precision and speed. In our scenario, each of the 8 cavities represents a different potential version of the part being produced. If the process is uniform, each cavity has an equal chance of being chosen.
Thus, understanding the distribution of these cavities within the injection molding practice is crucial to calculating probabilities accurately and ensuring quality control in manufacturing.

Cavity probability refers to the likelihood of a part being produced from a specific cavity in the injection mold. This probability depends on the uniformity and working conditions of the injection molding process. In our situation, as each cavity is equally likely to produce a part, we consider this uniform distribution.To calculate cavity probability, we apply the basic probability formula:

• Probability of a specific cavity = \( \frac{\text{Number of favorable outcomes}}{\text{Total outcomes in sample space}} \)

For instance, the probability that a part comes from cavity 1 or 2 (favorable outcomes \( \{1, 2\} \)) is \( \frac{2}{8} = \frac{1}{4} \).
Similarly, calculating the probability for parts from neither cavity 3 nor 4 involves non-favorable outcomes \( \{1, 2, 5, 6, 7, 8\} \), resulting in a probability of \( \frac{6}{8} = \frac{3}{4} \).
Understanding these probabilities is important for analyzing production trends and ensuring consistency.