Let's solve the given questions step by step.
### Question (i): The range of the data, 15, 4, 16, 20, 5, 6, 16, 8, 2, 1, 19, 0
**Step 1**: Identify the highest and lowest values in the data set.
- The highest value is 20.
- The lowest value is 0.
**Step 2**: Calculate the range using the formula:
\[
\text{Range} = \text{Highest Value} - \text{Lowest Value}
\]
\[
\text{Range} = 20 - 0 = 20
\]
**Answer**: The range of the data is **20**. Therefore, \( ul(P) = 20 \).
### Question (ii): Probability of an impossible event
**Step 1**: Understand the definition of an impossible event.
- An impossible event is one that cannot occur at all.
**Step 2**: Determine the probability of an impossible event.
- The probability of an impossible event is always **0**.
**Answer**: The probability of an impossible event is **0**. Therefore, \( ul(Q) = 0 \).
### Question (iii): The number of times a particular observation occurs in given data
**Step 1**: Understand the term used for the frequency of an observation.
- The number of times a particular observation occurs is known as its **frequency**.
**Answer**: The number of times a particular observation occurs is called **frequency**. Therefore, \( ul(R) = \text{frequency} \).
### Question (iv): In a single throw of two dice, the probability of getting a total of 11
**Step 1**: Identify the combinations that give a total of 11 when two dice are thrown.
- The combinations are (5, 6) and (6, 5).
**Step 2**: Count the total number of possible outcomes when throwing two dice.
- The total outcomes = 6 (for the first die) × 6 (for the second die) = 36.
**Step 3**: Count the favorable outcomes for getting a total of 11.
- There are 2 favorable outcomes: (5, 6) and (6, 5).
**Step 4**: Calculate the probability using the formula:
\[
\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{2}{36} = \frac{1}{18}
\]
**Answer**: The probability of getting a total of 11 is \( \frac{1}{18} \). Therefore, \( ul(S) = \frac{1}{18} \).
### Summary of Answers:
- \( ul(P) = 20 \)
- \( ul(Q) = 0 \)
- \( ul(R) = \text{frequency} \)
- \( ul(S) = \frac{1}{18} \)
