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In the realm of probability, the concept of a "sample space" is fundamental. Imagine Nick flipping a coin three times. Each flip is simple, either landing on heads (H) or tails (T). However, when combined, the outcomes become numerous.

The sample space is the complete set of all possible results from an experiment, outlining every potential combination from the flips. With each coin flip having 2 possible results, if we multiply these by themselves three times (once for each flip), that's a total of 8 combinations. Therefore, the sample space for Nick’s experiment is given by \[S = \{HHH, HHT, HTH, THH, TTH, THT, HTT, TTT\}\].

Each sequence in this set represents a unique way the coins could land over the course of the three flips. Knowing this sample space is crucial as it lays the groundwork for calculating probabilities of specific events.

An "event" in probability is a subset of the sample space. It refers to the occurrence we are interested in, like a specific combination of coin toss results. For Nick's coin flipping activity, a particular event could be where heads show up more often than tails.

To define this, we want scenarios where there are either two heads and one tail or all three heads in the sequence. Referring back to the sample space:

• 3 heads: HHH
• 2 heads and 1 tail: HHT, HTH, THH

Thus, our event consists of these outcomes: \[E = \{HHH, HHT, HTH, THH\}\].

By understanding events, we can focus on particular sequences that interest us in probability, helping us calculate the likelihood of specific outcomes.

A "coin flip experiment" is a classic probability exercise because it's easy to visualize and statistically fair – assuming the coin is unbiased. With each coin flip, there are two possible outcomes: heads (H) or tails (T). This simplicity makes it an excellent model for studying basic probability.

In Nick's experiment, he flips a coin three times. The task involves calculating the probabilities of these flips, starting with identifying the complete sample space and then honing in on particular events, such as heads appearing more often than tails.

Coin flip experiments are foundational to understanding probability because they help illustrate the concepts of random processes and outcomes:

• Random events with equal probabilities
• Accumulating probability over multiple trials
• Using outcomes to determine probabilities of events