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A dartboard has 5 equally sized slices numbered from 1 to 5.
Some are grey and some are white.
The slices numbered 3 and 4 are grey.
The slices numbered 1,2 , and 5 are white.
A dart is tossed and lands on a slice at random.
Let X be the event that the dart lands on a grey slice, and let P(X) be the probability of X.
Let not X be the event that the dart lands on a slice that is not grey, and let P(not X) be the probability of not X. (a)For each event in the table, check the outcome(s) that are contained in the event. Then, in the last column, enter the probability of the event.

Let's solve this probability problem step by step.

### Step 1: Identify the Outcomes for Events \( X \) and \( \text{not } X \)
- **Event \( X \):** The dart lands on a grey slice.
  - Grey slices are numbered 3 and 4.
- **Event \( \text{not } X \):** The dart lands on a slice that is not grey.
  - Non-grey (white) slices are numbered 1, 2, and 5.

### Step 2: Mark the Outcomes in the Table
- For **Event \( X \)**, mark slices 3 and 4.
- For **Event \( \text{not } X \)**, mark slices 1, 2, and 5.

### Step 3: Calculate the Probabilities
- **Probability of \( X \) (P(X)):**
  - There are 5 possible outcomes (slices 1 to 5).
  - The favorable outcomes for \( X \) are 2 (slices 3 and 4).
  - So, \( P(X) = \frac{2}{5} = 0.4 \).

- **Probability of \( \text{not } X \) (P(not X)):**
  - The favorable outcomes for \( \text{not } X \) are 3 (slices 1, 2, and 5).
  - So, \( P(\text{not } X) = \frac{3}{5} = 0.6 \).

### Step 4: Fill in the Table
- **Event \( X \):** Outcomes 3 and 4 checked, \( P(X) = 0.4 \).
- **Event \( \text{not } X \):** Outcomes 1, 2, and 5 checked, \( P(\text{not } X) = 0.6 \).

Here’s how the completed table should look:

\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\textbf{Event} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{Probability} \\
\hline
X &  &  & \checked & \checked &  & 0.4 \\
\hline
\text{not } X & \checked & \checked &  &  & \checked & 0.6 \\
\hline
\end{array}
\]

Would you like more details on how these probabilities were calculated or have any other questions?

### Relative Questions:
1. How do you calculate the probability of an event in a uniform sample space?
2. Can a probability be greater than 1 or less than 0? Why or why not?
3. What is the sum of the probabilities of all possible outcomes in a sample space?
4. How would the probabilities change if the dartboard had different sizes of slices?
5. What is the difference between mutually exclusive and independent events?
6. Can an event and its complement have the same probability? Under what condition?
7. How do probabilities help in making decisions under uncertainty?
8. What are the real-life applications of complement events in probability?

### Tip:
Always remember that the sum of the probabilities of an event and its complement is always 1.

Learn how to calculate the probability of a dart landing on grey slices on a dartboard with specific configurations. Understand the concepts of probability events and complements. Suitable for high school students studying probability theory.

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