A dartboard has 10 equally sized slices numbered from 1 to 10.
Some are grey and some are white.
The slices numbered 1,7 , and 9 are grey.
The slices numbered 2,3 ,4 ,5 ,6 ,8 , and 10 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 break down the problem and calculate the probabilities.

### Step 1: Identify the possible outcomes

There are 10 equally sized slices on the dartboard, numbered from 1 to 10. These slices are divided into grey and white:

- **Grey slices**: 1, 7, 9
- **White slices**: 2, 3, 4, 5, 6, 8, 10

### Step 2: Define the events

- **Event X**: The dart lands on a grey slice.
- **Event not X**: The dart lands on a slice that is not grey (i.e., a white slice).

### Step 3: List the outcomes for each event

- **Outcomes for X**: The dart lands on slices 1, 7, or 9.
- **Outcomes for not X**: The dart lands on slices 2, 3, 4, 5, 6, 8, or 10.

### Step 4: Calculate the probabilities

- **Total possible outcomes**: 10 (since there are 10 slices).
- **Number of outcomes favorable to X**: 3 (slices 1, 7, 9).
- **Number of outcomes favorable to not X**: 7 (slices 2, 3, 4, 5, 6, 8, 10).

The probability of an event is calculated as:

\[
P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}
\]

- **Probability of X**: 
\[
P(X) = \frac{3}{10} = 0.3
\]
- **Probability of not X**: 
\[
P(\text{not } X) = \frac{7}{10} = 0.7
\]

### Step 5: Fill in the table

Based on the explanation above, here’s how the table would look:

| **Event**      | **Outcome(s) contained in the event** | **Probability of the event** |
|----------------|---------------------------------------|------------------------------|
| **X**          | 1, 7, 9                               | 0.3                          |
| **not X**      | 2, 3, 4, 5, 6, 8, 10                  | 0.7                          |

Would you like further details or explanations on this? If not, here are some related questions you might find interesting:

1. What is the probability of hitting a white slice?
2. If the dartboard had 12 slices, how would the probabilities change?
3. How would the probabilities change if another grey slice were added?
4. What is the probability that the dart lands on an even-numbered slice?
5. How would you calculate the probability if the slices were not equally sized?
6. Can you explain the difference between independent and dependent events in probability?
7. How does the probability change if the dartboard is biased?
8. How would you apply the concept of conditional probability to this scenario?

**Tip:** When calculating probabilities, always ensure that the events are mutually exclusive and collectively exhaustive for a complete analysis.

Understand the probability of a dart landing on grey or white slices on a dartboard. Calculate the probabilities for events X and not X, where X is landing on a grey slice and not X is landing on a white slice. Suitable for students in Grades 7-9, this problem explores basic concepts of probability theory.

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