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In probability theory, the sample space refers to the set of all possible outcomes of a particular experiment. When we're considering an experiment like tossing two balanced coins, we're looking at all the different ways the coins can land. Each possible combination of outcomes is called a sample point.
For our coin-tossing experiment, the sample space includes these possibilities:

• Both coins show heads (HH)
• The first coin shows heads while the second shows tails (HT)
• The first coin shows tails while the second shows heads (TH)
• Both coins show tails (TT)

Understanding the sample space is a crucial first step in calculating probabilities, as it represents the full set of results that we will evaluate.

Probability helps us find the likelihood of specific events occurring. In our coin-tossing scenario, when the coins are balanced, each outcome is equally likely.
With four possible outcomes (HH, HT, TH, TT) and equal probability distribution, each sample point has a probability of occurring of \( \frac{1}{4} \). This is because there are four total sample points, and the probability of each event happening (assuming all are equally likely) is the inverse of the total number of outcomes.
In terms of events such as "exactly one head," or "at least one head," we express the event probability as a fraction of the sample points. The probability of event \(A\) (exactly one head) is computed by considering the sample points that meet this condition: HT and TH. Hence, \(P(A) = \frac{2}{4} = \frac{1}{2}\). Similarly, the probability of event \(B\) (at least one head) is derived from HT, TH, and HH, resulting in \(P(B) = \frac{3}{4}\). This segmentation helps in analyzing events based on their specific characteristics.

Events are considered independent when the occurrence of one event does not affect the probability of the other. In this coin-tossing scenario, the toss of one coin does not impact the toss of the other coin. This means each coin toss is an independent event.
The probability of flipping one head or tail remains \( \frac{1}{2} \), regardless of previous or concurrent outcomes. Recognizing events as independent simplifies the calculation of joint probabilities, as individual probabilities can be multiplied. Even when calculating \(P(A \cap B)\) or the probability of intersections in independent events, the independence allows treating each event distinctly while preserving their unique likelihoods.

Combinatorics is a branch of mathematics that studies the combination of objects. It helps us count different possible outcomes and solve problems related to probability. In our coin-tossing example, we use combinatorics to determine all potential outcomes.
The basic combinatorial method is listing possibilities or arranging, depending on the number of objects. Here, the objects are two coins, and each coin has two positions, resulting in © combinations. This results in the four outcomes we have analyzed as our sample space: HH, HT, TH, and TT.
Combinatorics also lends itself to formulating probabilities efficiently when more complex situations arise, assisting in calculating and analyzing interesting events and outcomes systematically.