Let's address problem 50 from the uploaded image.

Problem 50

It asks us to calculate the expected value and standard deviation for a lottery game called "Tri-State Pick 3". Here's the setup:

1. You buy one Pick 3 ticket for two consecutive days.
2. For each day:
   • The winnings depend on the probability distribution provided.
   • The winnings from the two days are independent.

Let's calculate:

Step 1: Expected Value for One Day

For a single day: $E(X) = \sum (\text{Value} \cdot \text{Probability})$

From the table (if the distribution is not visible, let me know for clarification):

• Suppose winnings values are $x_1, x_2, \ldots, x_n$ with corresponding probabilities $p_1, p_2, \ldots, p_n$.

Substitute and compute $E(X)$.

Step 2: Variance for One Day

Variance is given by: $\text{Var}(X) = \sum [(\text{Value} - E(X))^2 \cdot \text{Probability}]$

Then, compute $\text{Var}(X)$ and its square root for the standard deviation.

Step 3: Two-Day Winnings

Since the winnings for two days are independent, the total expected value is: $E(\text{Total}) = 2 \cdot E(X)$

And the total variance is: $\text{Var}(\text{Total}) = 2 \cdot \text{Var}(X)$

Finally, the standard deviation for the total is: $\text{SD}(\text{Total}) = \sqrt{\text{Var}(\text{Total})}$

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Would you like me to compute this explicitly with your data or clarify anything? Here are five questions based on this problem:

1. What is the formula for calculating expected value for a discrete random variable?
2. How does independence of random variables affect the computation of variances?
3. What would happen to the expected value if you played for 10 days instead of 2?
4. Why is it important to know both the expected value and the standard deviation for gambling scenarios?
5. How would the calculations change if the winnings on one day affected the next day’s winnings?

Tip: Always verify that the probabilities sum to 1 before performing calculations.