A probability experiment is conducted in which the sample space of the experiment is $$ \mathrm{S}=\{1,2,3,4,5,6,7,8,9,10,11,12\} $$, event $$ F=\{5,6,7,8,9,10\} $$, and event $$ \mathrm{G}=\{9,10,11,12\} $$. Assume that each outcome is equally likely. List the outcomes in F or G . Find $$ \mathrm{P}(\mathrm{F} $$ or G$$ ) $$ by counting the number of outcomes in F or G . Determine $$ \mathrm{P}(\mathrm{F} $$ or G$$ ) $$ using the general addition rule. List the outcomes in $$ F $$ or G . Select the correct choice below and, if necessary, fill in the answer box to complete your choice. - A. F or $$ \mathrm{G}=\{5,6,7,8,9,10,11,12\} $$ (Use a comma to separate answers as needed.) B. F or $$ \mathrm{G}=\{ \} $$ Find $$ P(F $$ or $$ G) $$ by counting the number of outcomes in $$ F $$ or $$ G $$. $$ P(F $$ or $$ G)=\square $$ (Type an integer or a decimal rounded to three decimal places as needed.)

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**Step 1. List the outcomes in $$ F $$ and $$ G $$.**

$$
F = \{5, 6, 7, 8, 9, 10\}, \quad G = \{9, 10, 11, 12\}
$$

**Step 2. Determine the union $$ F $$ or $$ G $$ (i.e. $$ F \cup G $$).**

The union $$ F \cup G $$ includes all outcomes that are in $$ F $$ or $$ G $$:

$$
F \cup G = \{5, 6, 7, 8, 9, 10, 11, 12\}
$$

This corresponds to option A.

**Step 3. Count the outcomes in $$ F \cup G $$.**

There are $$8$$ outcomes in $$ F \cup G $$.

**Step 4. Compute $$ P(F $$ or $$ G) $$ by counting the outcomes.**

Since each outcome in the sample space $$ S = \{1,2,\dots,12\} $$ is equally likely, the probability is:

$$
P(F \text{ or } G) = \frac{\text{Number of outcomes in } F \cup G}{\text{Total number of outcomes in } S} = \frac{8}{12} = \frac{2}{3} \approx 0.667
$$

**Step 5. Verify $$ P(F $$ or $$ G) $$ using the general addition rule.**

The general addition rule is:

$$
P(F \text{ or } G) = P(F) + P(G) - P(F \cap G)
$$

First, compute:

- $$P(F) = \frac{6}{12} = 0.5$$ (since $$ F $$ has 6 outcomes),
- $$P(G) = \frac{4}{12} \approx 0.333$$ (since $$ G $$ has 4 outcomes).

The intersection $$ F \cap G $$ is:

$$
F \cap G = \{9, 10\}
$$

Thus,

$$
P(F \cap G) = \frac{2}{12} \approx 0.167
$$

Now, use the rule:

$$
P(F \text{ or } G) = 0.5 + 0.333 - 0.167 \approx 0.666 \approx 0.667
$$

**Final Answers:**

- **A. $$ F $$ or $$ G = \{5, 6, 7, 8, 9, 10, 11, 12\} $$**
- **$$ P(F $$ or $$ G) = 0.667 $$ (or equivalently, $$ \frac{2}{3} $$)**
**Outcomes in $$ F $$ or $$ G $$:** 5, 6, 7, 8, 9, 10, 11, 12 **Probability $$ P(F $$ or $$ G) $$:** 0.667

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