Question 1190338: Ten thousand lottery tickets are sold at $10 each. First prize is $10000. There are 3 second prizes at $5000 each. There are 10 third prizes at $1000 each. If X is the discrete random variable that measures the gain from buying a lottery ticket, then
a. What is the probability distribution of X?
b. What is the expected value of X?
c. What is the standard deviation of X?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! 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.
|
|
|
