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Let's break down the problem from the image. We are working with a fair coin and a fair die, and the sample space provided is:

\[
S = \{ H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6 \}
\]

Each outcome consists of a coin flip (H for heads, T for tails) and a die roll (1 through 6).

### Definitions:

- \( A \): Heads is flipped (i.e., outcomes \( H1, H2, H3, H4, H5, H6 \))
- \( B \): Tails is flipped (i.e., outcomes \( T1, T2, T3, T4, T5, T6 \))
- \( C \): An even number is rolled (i.e., outcomes \( H2, H4, H6, T2, T4, T6 \))
- \( D \): A number less than 3 is rolled (i.e., outcomes \( H1, H2, T1, T2 \))

We are asked to calculate:

### a) \( P(A) \)
- \( A \) represents flipping a head. There are 6 outcomes where heads is flipped: \( \{ H1, H2, H3, H4, H5, H6 \} \).
- The total number of outcomes in the sample space is 12.
  
So, \( P(A) = \frac{6}{12} = \frac{1}{2} \).

### b) \( P(C^c \cup D) \)
- \( C^c \) is the complement of \( C \), meaning the roll is not an even number. The outcomes for odd numbers are \( H1, H3, H5, T1, T3, T5 \), which gives 6 outcomes.
- \( D \) is rolling a number less than 3, so the outcomes are \( H1, H2, T1, T2 \), which gives 4 outcomes.
- Now we compute \( C^c \cup D \), which includes all outcomes that are either not even or less than 3. These are \( H1, H3, H5, T1, T3, T5, H2, T2 \). The total distinct outcomes are 8.

So, \( P(C^c \cup D) = \frac{8}{12} = \frac{2}{3} \).

### c) \( P(D^c \cap A^c) \)
- \( D^c \) is the complement of \( D \), so the roll is not less than 3. The outcomes are \( H3, H4, H5, H6, T3, T4, T5, T6 \), which gives 8 outcomes.
- \( A^c \) means tails is flipped, i.e., outcomes are \( T1, T2, T3, T4, T5, T6 \), which gives 6 outcomes.
- Now we compute \( D^c \cap A^c \), which includes outcomes that are both not less than 3 and tails is flipped. These are \( T3, T4, T5, T6 \). The total outcomes are 4.

So, \( P(D^c \cap A^c) = \frac{4}{12} = \frac{1}{3} \).

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Would you like further details or clarification on any part of this solution? Here are 5 related questions you can explore:

1. How would the probabilities change if we used a biased coin instead of a fair one?
2. What is the probability of rolling a number greater than 4 and flipping tails?
3. Calculate \( P(A \cap C) \), the probability of flipping heads and rolling an even number.
4. How does the complement of an event affect its probability?
5. What is the probability of rolling a number greater than 2 and flipping heads?

**Tip:** When calculating probabilities involving multiple events, it is helpful to clearly distinguish between unions (\( \cup \)) and intersections (\( \cap \)) to avoid errors.

A fair coin is flipped and a fair die is rolled. The sample space is S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}. Each outcome is equally likely to occur. Define A = heads is flipped, B = tails is flipped, C = an even number is rolled, D = a number less than 3 is rolled. Calculate: a) P(A), b) P(C^c ∪ D), c) P(D^c ∩ A^c).

Explore the probability of flipping a coin and rolling a die with detailed step-by-step calculations for P(A), P(C^c ∪ D), and P(D^c ∩ A^c). This problem involves understanding complements, unions, and intersections of sets, making it ideal for high school students studying probability.

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