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In probability, the concept of a sample space refers to the set of all possible outcomes of a particular experiment or random process. Think of it as the complete list of every event that could happen when performing an activity. In our exercise, the sample space is determined by the act of selecting three coins from a jar containing a nickel, a dime, a quarter, and a half-dollar.
To find the sample space, you list all permutations of the activity. For the exercise at hand, the problem involves selecting coins. By examining the different combinations, we find the following simple events or outcomes in the sample space:

• Nickel, Dime, Quarter (NDQ)
• Nickel, Dime, Half-Dollar (NDH)
• Nickel, Quarter, Half-Dollar (NQH)
• Dime, Quarter, Half-Dollar (DQH)

These four outcomes represent the complete sample space, as they cover all potential groupings of three coins picked from the jar.

Favorable outcomes are the particular outcomes from the sample space that satisfy a given condition or event. In the context of our exercise, we see this in step three where we focus on selecting a half-dollar among the three coins.
To determine which outcomes involve the half-dollar, review the list of possible events:

• Nickel, Dime, Half-Dollar (NDH)
• Nickel, Quarter, Half-Dollar (NQH)
• Dime, Quarter, Half-Dollar (DQH)

Here, three of the four possible events include the half-dollar, making them our favorable outcomes. Whether figuring out if something specific is included (like the half-dollar), or achieving a particular condition (such as reaching a specific total value), favorable outcomes help narrow down and identify what fits within these criteria.

Probability calculation is the method of determining how likely it is for a particular outcome to occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
From our sample space and favorable outcomes, let's calculate the probability of selecting the half-dollar: The total number of outcomes is four, and the number of favorable outcomes where the selection includes the half-dollar is three. So, the probability is calculated as follows:\[P(\text{Half-Dollar}) = \frac{3}{4} = 0.75\]Thus, the probability of drawing a selection that includes the half-dollar from the jar of coins is 0.75, or 75%.In addition to merely selecting coins, probability calculation is also applied to situations where certain conditions must be met. For instance, finding the probability that the total value of selected coins is at least 60 cents follows similar steps: with the same number of favorable outcomes as before, we again use:\[P(\text{Total amount} \geq 60 \text{ cents}) = \frac{3}{4} = 0.75\]

Simple events represent the most basic possible outcomes of a probability experiment. They occur when a single event is considered without any additional conditions or combinations.
In our coin selection scenario, each combination of three selected coins forms a simple event. These simple events are specific and cannot be broken down into smaller outcomes. For instance:

• NDQ: Nickel, Dime, Quarter
• NDH: Nickel, Dime, Half-Dollar
• NQH: Nickel, Quarter, Half-Dollar
• DQH: Dime, Quarter, Half-Dollar

Understanding simple events helps in identifying the foundational outcomes within a sample space which, when combined, form more complex events or help in easily calculating probabilities by focusing on individual outcomes.