Question 1190338
Here's how to break down this lottery ticket problem:

**a. Probability Distribution of X (Gain):**

First, we need to define the possible values of X (the gain).  Remember, gain is what you win *minus* the cost of the ticket ($10).

*   **First Prize:** Gain = $10,000 - $10 = $9,990
*   **Second Prize:** Gain = $5,000 - $10 = $4,990
*   **Third Prize:** Gain = $1,000 - $10 = $990
*   **No Prize:** Gain = $0 - $10 = -$10

Now, let's calculate the probabilities of each outcome:

*   **P(X = $9,990):** 1 ticket out of 10,000 wins the first prize, so the probability is 1/10,000 = 0.0001
*   **P(X = $4,990):** 3 tickets out of 10,000 win a second prize, so the probability is 3/10,000 = 0.0003
*   **P(X = $990):** 10 tickets out of 10,000 win a third prize, so the probability is 10/10,000 = 0.001
*   **P(X = -$10):** The remaining tickets (10,000 - 1 - 3 - 10 = 9,986) win no prize, so the probability is 9,986/10,000 = 0.9986

Here's the probability distribution:

| Gain (X) | Probability P(X) |
|---|---|
| $9,990 | 0.0001 |
| $4,990 | 0.0003 |
| $990 | 0.001 |
| -$10 | 0.9986 |

**b. Expected Value of X:**

The expected value (E[X]) is calculated as:

E[X] = Σ [x * P(x)]  (summed over all possible values of x)

E[X] = ($9,990 * 0.0001) + ($4,990 * 0.0003) + ($990 * 0.001) + (-$10 * 0.9986)
E[X] = $0.999 + $1.497 + $0.99 - $9.986
E[X] = -$6.50

The expected value of X is -$6.50. This means that on average, you can expect to lose $6.50 for each lottery ticket you buy.

**c. Standard Deviation of X:**

1.  **Calculate E[X²]:**
    E[X²] = Σ [x² * P(x)]
    E[X²] = (9990² * 0.0001) + (4990² * 0.0003) + (990² * 0.001) + (-10² * 0.9986)
    E[X²] = 99800.1 + 7485.003 + 980.1 + 99.86
    E[X²] ≈ 108365.06

2.  **Calculate Variance (Var[X]):**
    Var[X] = E[X²] - (E[X])²
    Var[X] = 108365.06 - (-6.50)²
    Var[X] = 108365.06 - 42.25
    Var[X] ≈ 108322.81

3.  **Calculate Standard Deviation (SD[X]):**
    SD[X] = √Var[X]
    SD[X] = √108322.81
    SD[X] ≈ $329.12

The standard deviation of X is approximately $329.12.