91Ó°ÊÓ

In probability, the sample space is a fundamental concept that forms the basis of any probability experiment. It represents the set of all possible outcomes that can occur. For example, when dealing with injection-molded parts from a machine, each cavity from which a part could come is an individual outcome.

In our specific case, we have eight cavities, so the sample space is written as:

• \( S = \{1, 2, 3, 4, 5, 6, 7, 8\} \)

Each number represents one cavity where a part might originate. By listing all possible outcomes, we ensure a comprehensive understanding of the potential scenarios that our probability questions might ask about.

It's crucial to clearly define this sample space, as it serves as the foundation for other probability concepts like favorable and unfavorable outcomes.

Favorable outcomes are specific outcomes within the sample space that meet the criteria of the event we're interested in. Identifying these outcomes allows us to calculate the probability of the event occurring.

For instance, if we're concerned with the probability of a part being from cavity 1 or 2, these two outcomes (1 and 2) are our favorable outcomes. This is because they are directly linked to the condition we're examining.

• Favorable outcomes: 1, 2

To find the probability of a favorable outcome, we divide the number of favorable outcomes by the total number of possible outcomes (from the sample space). In this scenario, the probability is:

• \( P(\text{from cavity 1 or 2}) = \frac{2}{8} = \frac{1}{4} \)

By focusing on the favorable outcomes, we can clearly and accurately determine the likelihood of the event of interest.

Unfavorable outcomes are those within the sample space that do not satisfy the specific event's criteria. Recognizing these outcomes is essential when calculating the probability of the event not occurring or its complementary event.

Using our cavity example, if we want the probability that a part is from neither cavity 3 nor 4, the cavities 3 and 4 are unfavorable because they are explicitly excluded from the desired event.

• Unfavorable outcomes: 3, 4

This means that favorable outcomes are the rest of the sample space excluding these unfavorable ones.

• Favorable outcomes: 1, 2, 5, 6, 7, 8

Consequently, the probability of the part coming from a cavity other than 3 or 4 is calculated by dividing the number of favorable outcomes by the total outcomes:

• \( P(\text{neither 3 nor 4}) = \frac{6}{8} = \frac{3}{4} \)

Understanding the concept of unfavorable outcomes helps us complete the picture of probability by considering all scenarios that do not meet the event's conditions.