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In probability theory, a sample space is a fundamental concept. It refers to the set of all possible outcomes of an experiment or a probability event. In this context, the envelope problem provides a clear illustration of a sample space. Here, we are dealing with envelopes that contain different amounts of money. The possible outcomes, or the sample space, include \(100, \)25, and $10. These are the only different amounts one could possibly discover when opening any envelope from the box.

Sample space is crucial because it defines the scope of our probabilistic analysis. By identifying it, we know what outcomes we must consider when making calculations or predictions. In broader applications, sample spaces can be as simple as a toss of a coin, with outcomes \("Heads"\) or \("Tails"\), or as complex as the possible results of a sports tournament. By identifying and analyzing the sample space, one effectively sets the stage for solving probability exercises.

Understanding mutually exclusive events helps us in calculating probabilities accurately. In the scenario of determining the money inside the envelope, mutually exclusive events are those that cannot happen at the same time. When discussing envelopes containing money, a single envelope cannot contain both $25 and $10 at the same time. Thus, these are mutually exclusive events.

When events are mutually exclusive, the occurrence of one event means the other cannot occur. This characteristic is key when determining probabilities since it allows us to use the additive rule. Simply put, if we want to find the probability of one event or another mutually exclusive event occurring, we sum their individual probabilities. For instance, if we want to find the odds of picking an envelope that contains less than $100, we combine the probabilities of the $25 and $10 events. This provides a straightforward way to handle scenarios where multiple outcomes are possible but cannot occur simultaneously.

Assigning probabilities to events within our sample space allows us to understand how likely each outcome is. In the envelope problem, we know there are 500 envelopes in total, distributed across different values: 75 with \(100, 150 with \)25, and 275 with \(10.

• Probability of picking a \)100 envelope: \( \frac{75}{500} = 0.15 \)
• Probability of picking a \(25 envelope: \( \frac{150}{500} = 0.3 \)
• Probability of picking a \)10 envelope: \( \frac{275}{500} = 0.55 \)

Probabilities are essential for making informed predictions about which events are more likely to occur. In any given experiment or random process, every event within the sample space can have a probability assigned based on its frequency or likelihood of occurrence. These probabilities must sum to 1, as they represent the entirety of the sample space.

With this understanding, you can tackle numerous probability-based problems, utilizing the frequencies or occurrences of events to anticipate outcomes and make decisions accordingly.