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The given problem involves a finite sample space \( \Omega = \{0, 1, 2, \dots, n\} \) for a positive integer \( n \), and a probability function \( \mathbb{P}(\{k\}) = c k \) for each \( k \in \Omega \), where \( c \) is a constant to be determined. You are asked to:

1. **Find \( c \) in terms of \( n \)** (1 point).
2. **Verify that this satisfies the first and second axioms of probability** (2 points).

### Step 1: Finding \( c \) in terms of \( n \)

The sum of the probabilities over the entire sample space must be 1 (this is the **normalization condition**):

\[
\sum_{k=0}^{n} \mathbb{P}(\{k\}) = 1
\]

Since \( \mathbb{P}(\{k\}) = c k \), this gives:

\[
\sum_{k=0}^{n} c k = 1
\]

Factoring out \( c \):

\[
c \sum_{k=0}^{n} k = 1
\]

The sum of the first \( n \) integers is a well-known formula:

\[
\sum_{k=0}^{n} k = \frac{n(n+1)}{2}
\]

Substituting this into the equation:

\[
c \cdot \frac{n(n+1)}{2} = 1
\]

Solving for \( c \):

\[
c = \frac{2}{n(n+1)}
\]

Thus, the constant \( c \) is:

\[
c = \frac{2}{n(n+1)}
\]

### Step 2: Verifying the Axioms of Probability

The two fundamental axioms of probability are:

- **Axiom 1**: The probability of any event is non-negative, i.e., \( \mathbb{P}(\{k\}) \geq 0 \) for all \( k \in \Omega \).
- **Axiom 2**: The sum of the probabilities of all possible outcomes is 1, i.e., \( \sum_{k=0}^{n} \mathbb{P}(\{k\}) = 1 \).

#### Verifying Axiom 1

Since \( \mathbb{P}(\{k\}) = c k \), and \( k \geq 0 \), the probability will always be non-negative as long as \( c \geq 0 \). From our earlier result, \( c = \frac{2}{n(n+1)} \), which is positive for any positive integer \( n \). Therefore, the first axiom is satisfied.

#### Verifying Axiom 2

We already used the second axiom in Step 1 when we summed the probabilities and set it equal to 1. Thus, the second axiom is also satisfied by the calculation:

\[
\sum_{k=0}^{n} \mathbb{P}(\{k\}) = 1
\]

### Conclusion

- The constant \( c \) is \( \frac{2}{n(n+1)} \).
- The probability function satisfies both the first and second axioms of probability.

Would you like further clarification or additional details?

Here are five related questions to deepen your understanding:
1. How would the result change if the sample space were modified?
2. What are the implications if \( c \) were not positive?
3. Can this probability distribution be generalized to non-uniform cases?
4. How would you handle the case when \( n \to \infty \)?
5. What if the probability \( \mathbb{P}(\{k\}) \) had a different dependence on \( k \)?

**Tip**: Always check that probability distributions sum to 1, as this is a fundamental requirement in probability theory.

Suppose sample space Ω = {0, 1, 2, ..., n} for some positive integer n. Let P({k}) = ck for all k ∈ Ω. Find c in terms of n (1 point) and verify that it satisfies the first and second axioms of probability (2 points).

Explore how to find the constant c in terms of n for the probability function P({k}) = ck over a finite sample space Ω = {0, 1, 2, ..., n}. Verify that the result satisfies the first and second axioms of probability. Suitable for undergraduate students studying probability theory.

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