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A sample space is a fundamental concept in probability theory. It is a set which includes all possible outcomes of a random experiment. In this case, our experiment involves tossing two nickels and one dime.
The possible outcomes for each coin are either heads (H) or tails (T). When combining all three coins, we have:

• Nickel 1: H or T
• Nickel 2: H or T
• Dime: H or T

Thus, our sample space consists of all combinations of these outcomes. This forms the set:

• \[S_1 = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}\]

The sample space \(S_1\) includes 8 outcomes because each coin can land in 2 ways, and \(2 \times 2 \times 2 = 8\). Every time you shake and throw these coins, the outcome will be one of these combinations.

A probability density function (PDF) describes the likelihood of each outcome in the sample space. In our coin tossing example, since each outcome is equally probable and the coins are fair, we describe this likelihood as follows:
For the sample space \( S_1\), each of the 8 potential outcomes occurs with the equal probability.

• Probability of each outcome \( P(x) = \frac{1}{8} \)

This probability reflects the fact that each sequence of heads and tails is as likely as any other.
Additionally, when considering a desired event like the number of heads (which determines the coins we "keep"), we define another sample space \(S_2\).Here,

• Outcomes include keeping 0, 1, 2, or 3 heads (\{0H, 1H, 2H, 3H\}).
• Corresponding PDF: \( P(0H) = \frac{1}{8}, P(1H) = \frac{3}{8}, P(2H) = \frac{3}{8}, \text{and } P(3H) = \frac{1}{8} \)

This PDF reflects the probabilities based on how many heads appear, considering the fairness and independence of each coin toss.

In probability and statistics, events are said to be independent when the occurrence of one event does not influence the outcome of another. This is particularly relevant in our exercise, where the coins tossed are fair and independent of each other.
For instance, the outcome of nickel 1 landing on heads or tails does not affect the outcome of nickel 2, nor does it affect the outcome of the dime. Because of this independence, the joint probability of any specific sequence of heads and tails is simply the product of their individual probabilities.

• Probability of heads on one coin \( = \frac{1}{2} \)
• Therefore, probability of all three coins showing heads \( (HHH) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \)

The concept of independent events is crucial in constructing sample spaces and calculating probability densities, allowing us to analyze how each toss independently contributes to the whole probability.