To solve the problem, we need to find the probability that either the die will come up 2 or 3, or the coin will land heads up. We will use the basic principles of probability to find the solution.
### Step-by-Step Solution:
1. **Identify the Sample Space**:
- When a fair die is rolled, the possible outcomes are: {1, 2, 3, 4, 5, 6}. So, there are 6 outcomes for the die.
- When a fair coin is flipped, the possible outcomes are: {Heads, Tails}. So, there are 2 outcomes for the coin.
- The total number of outcomes when both the die is rolled and the coin is flipped is:
\[
6 \times 2 = 12.
\]
2. **Define the Events**:
- Let \( A \) be the event that the die shows either 2 or 3.
- Let \( B \) be the event that the coin shows heads.
- We need to find \( P(A \cup B) \), the probability that either event \( A \) or event \( B \) occurs.
3. **Calculate \( P(A) \)**:
- The favorable outcomes for event \( A \) (die shows 2 or 3) are: {2, 3}. Thus, there are 2 favorable outcomes.
- Therefore, the probability of event \( A \) is:
\[
P(A) = \frac{\text{Number of favorable outcomes for A}}{\text{Total outcomes}} = \frac{2}{6} = \frac{1}{3}.
\]
4. **Calculate \( P(B) \)**:
- The favorable outcome for event \( B \) (coin shows heads) is: {Heads}. Thus, there is 1 favorable outcome.
- Therefore, the probability of event \( B \) is:
\[
P(B) = \frac{\text{Number of favorable outcomes for B}}{\text{Total outcomes}} = \frac{1}{2}.
\]
5. **Calculate \( P(A \cap B) \)**:
- Events \( A \) and \( B \) are independent (the outcome of the die does not affect the outcome of the coin).
- Therefore, the probability of both events occurring (i.e., the die shows 2 or 3 and the coin shows heads) is:
\[
P(A \cap B) = P(A) \times P(B) = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}.
\]
6. **Use the Formula for the Union of Two Events**:
- The formula for the probability of the union of two events is:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B).
\]
- Substituting the values we calculated:
\[
P(A \cup B) = \frac{1}{3} + \frac{1}{2} - \frac{1}{6}.
\]
7. **Finding a Common Denominator**:
- The common denominator for 3, 2, and 6 is 6.
- Convert each fraction:
\[
P(A \cup B) = \frac{2}{6} + \frac{3}{6} - \frac{1}{6} = \frac{2 + 3 - 1}{6} = \frac{4}{6} = \frac{2}{3}.
\]
8. **Final Answer**:
- The probability that either the die will come up 2 or 3, or the coin will land heads up is:
\[
\frac{2}{3}.
\]
