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In probability, the concept of a sample space is foundational. Think of it as the universe of all possible outcomes for any given experiment or event. For example, if you’re tossing a coin, the sample space is {Heads, Tails}. It encompasses every potential result you might observe. In our exercise about repairing shop components, the sample space includes all possible types of components, both audio and video, that could be brought into the shop. It visually represents every potential scenario the shop may face regarding incoming repairs.

• A sample space gives a complete set of all things that could happen.
• Each outcome in the sample space is unique and exclusive.

Understanding the sample space helps us recognize the events, such as components being audio or video, within the complete range of possibilities. This recognition is crucial when we dive into specific probabilities tied to these events.

Event probability measures the likelihood of a certain outcome within a sample space. For instance, when we talk about something happening, like it raining tomorrow or pulling a red card from a deck, we're discussing event probability. In the context of our exercise, the event probability helps determine how likely it is for a component brought in for repair to be audio, denoted as event \(A\), or a compact disc player, denoted as event \(B\). Given the probabilities, \(P(A) = 0.6\) and \(P(B) = 0.05\), we can make informed predictions about these occurrences.

• Event probability ranges from 0 to 1, where 0 means impossible and 1 means certain.
• Having the probability of each event helps in evaluating how often we can expect it to occur.

By knowing the probabilities of events \(A\) and \(B\), we can calculate their interaction, such as the probability of \(B\) occurring given that \(A\) has occurred.

The intersection of events considers how two events might overlap or occur together within a sample space. We denote this as \(A \cap B\), meaning both event \(A\) and event \(B\) happen at the same time. In the exercise, we see that event \(B\) is actually contained within event \(A\), meaning every time an event \(B\) occurs, event \(A\) happens as well. Thus, \(P(B \cap A)\) is the same as \(P(B)\). Such intersections help in calculating conditional probability using the formula \(P(B \mid A) = \frac{P(B \cap A)}{P(A)}\).

• An intersection narrows down the sample space to outcomes shared by both events.
• The overlap represents joint occurrences within the broader sample space.

Understanding these intersections makes it easier to assess combined probabilities and predict scenarios where multiple conditions hold true simultaneously.