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In probability theory, the sample space represents all possible outcomes of an experiment. Think of it as a complete list of everything that could happen once you perform the experiment. For our scenario, where we toss two coins (A and B), each coin can land either heads (H) or tails (T). Hence, our sample space is given by the set \( S = \{HH, HT, TH, TT\} \). Each element in this set corresponds to a specific combination of heads and tails for the two coins, providing us with a framework to determine probabilities for various events.

• **HH**: Both coins land heads.
• **HT**: Coin A is heads, coin B is tails.
• **TH**: Coin A is tails, coin B is heads.
• **TT**: Both coins land tails.

By establishing this sample space, we set the stage for calculating further probabilities, such as the likelihood of specific outcomes occurring.

Conditional probability is a key concept in probability theory. It refers to the probability of an event occurring given that another event has already occurred. This is vital for making predictions and understanding relationships between events.

In our exercise, we calculate certain probabilities based on conditions. For example, the probability that both coins will show heads given that coin A shows a head is a conditional probability task. We denote this as \( P(\text{HH} | H_A) \) and compute it using the formula:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Where \( P(A \cap B) \) is the probability of both events A and B occurring, and \( P(B) \) is the probability of event B. In our case, this formula helps us establish how the occurrence of one event (coin A landing heads) impacts another (both coins landing heads).

We used this method in various parts of the solution, ensuring we correctly considered the provided conditions.

A probability measure assigns a numeric probability to each possible outcome in a sample space. This assignment must satisfy certain criteria: each probability must be between 0 and 1, and the sum of all probabilities for the sample space must equal 1. In our example, we determine the probability measure by looking at the behavior of each coin.

For coin A, which is loaded, the probability of landing heads, \( P(H_A) \), is 0.6, while the probability of tails, \( P(T_A) \), is 0.4. For the balanced coin B, both heads and tails have equal probabilities of 0.5.

Using these probabilities, we assign a measure to each sample space outcome:

• **HH**: \( P(HH) = P(H_A) \times P(H_B) = 0.6 \times 0.5 = 0.3 \)
• **HT**: \( P(HT) = P(H_A) \times P(T_B) = 0.6 \times 0.5 = 0.3 \)
• **TH**: \( P(TH) = P(T_A) \times P(H_B) = 0.4 \times 0.5 = 0.2 \)
• **TT**: \( P(TT) = P(T_A) \times P(T_B) = 0.4 \times 0.5 = 0.2 \)

Through this method, we ensure that our probability assignments are accurate, forming the foundation for solving further probability questions related to the two coins.