Question 1193726
**1. Determine the Probability Mass Function (PMF) for X (Number of $2 Tickets)**

* **Total Tickets:** 3 (no win) + 3 (win $1) + 6 (win $2) = 12 tickets
* **Probability of winning $2:** 6/12 = 1/2 
* **Probability of not winning $2:** 6/12 = 1/2

* **Use the Binomial Probability Mass Function:** 
    * P(X = k) = (nCk) * p^k * (1-p)^(n-k) 
        * where:
            * n = number of trials (3 tickets)
            * k = number of successes (number of $2 tickets)
            * p = probability of success (probability of drawing a $2 ticket = 1/2)
            * nCk = binomial coefficient (number of combinations of n items taken k at a time) 

    * P(X = 0) = (3C0) * (1/2)^0 * (1/2)^3 = 1/8 
    * P(X = 1) = (3C1) * (1/2)^1 * (1/2)^2 = 3/8 
    * P(X = 2) = (3C2) * (1/2)^2 * (1/2)^1 = 3/8 
    * P(X = 3) = (3C3) * (1/2)^3 * (1/2)^0 = 1/8 

**2. Calculate the Average (Expected Value) of X**

* E(X) = Σ [k * P(X = k)] 
* E(X) = 0 * (1/8) + 1 * (3/8) + 2 * (3/8) + 3 * (1/8) 
* E(X) = (3 + 6 + 3) / 8 
* **E(X) = 1.5**

**3. Calculate the Variance of X**

* Var(X) = Σ [(k - E(X))² * P(X = k)]
* Var(X) = (0 - 1.5)² * (1/8) + (1 - 1.5)² * (3/8) + (2 - 1.5)² * (3/8) + (3 - 1.5)² * (1/8) 
* Var(X) = 2.25/8 + 0.25 * (3/8) + 0.25 * (3/8) + 2.25/8 
* Var(X) = 0.75

* **Standard Deviation (σ_X) = √Var(X) = √0.75 ≈ 0.866**

**4. Calculate the Average (Expected Value) of S**

* **Possible Winnings (S):**
    * 0 euros (no winning tickets)
    * 1 euro (one 1-euro ticket, no 2-euro tickets)
    * 2 euros (two 1-euro tickets, no 2-euro tickets or one 1-euro and one 2-euro ticket)
    * 3 euros (three 1-euro tickets or one 1-euro and one 2-euro ticket)
    * 4 euros (two 2-euro tickets and one 1-euro ticket)
    * 6 euros (three 2-euro tickets)

* Calculate the probability of each winning scenario (this can be more complex).

* **E(S) = Σ [s * P(S = s)]** 

* **Note:** Calculating E(S) and its variance would require a more detailed analysis of all possible winning scenarios and their probabilities.

**5. Correlation between X and S**

* There is a clear relationship between X (number of $2 tickets) and S (total winnings). 
* As X increases, the total winnings (S) generally increase. 
* Therefore, we would expect a **positive correlation** between X and S.

**6. Calculate the Correlation Coefficient (ρ)**

* Calculating the correlation coefficient (ρ) between X and S would require:
    * The joint probability distribution of X and S.
    * Calculating the covariance between X and S.
    * Calculating the standard deviation of S (σ_S).

* **ρ = Cov(X, S) / (σ_X * σ_S)**

**Key Considerations:**

* This analysis provides a general framework.
* Calculating the exact probabilities and expected values for S can be more involved.
* Using statistical software or programming tools can help with the calculations and simulations.

This analysis provides a foundation for understanding the relationship between the number of $2 winning tickets (X) and the total winnings (S) in this lottery scenario.