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The expected value, often referred to as the mean, is a fundamental concept in probability and statistics that provides a single summary number representing the center or "average" outcome of a probability distribution. It is useful for indicating what one can anticipate over a large number of trials.
In the context of a lottery, such as the Tri-State Pick 3, the expected value helps players understand the average payoff they can expect if they purchase many tickets.
In this game, the expected value of winnings, represented as \( \mu_X \), is computed by multiplying each possible payoff by its probability and summing these products. The expected value equation is:
\[ \mu_X = \sum (\text{value} \times \text{probability}) \]
Applied to our example:

• The payoff of \(\\(0\) with a probability of 0.999 contributes \(0 \times 0.999 = 0 \).
• The payoff of \(\\)500\) with a probability of 0.001 contributes \(500 \times 0.001 = 0.5 \).

Summing these gives the expected value \(\mu_X = \\(0.50\).
Thus, on average, a player can expect to win \(\\)0.50\) for each ticket purchased, even though the actual result of each ticket is typically either \(\\(0\) or \(\\)500\).
This information is crucial as it frames lottery games as instances of financial loss, further analyzed with different metrics such as variance and standard deviation.

Variance provides a measure of how much the values of a probability distribution spread out from the mean. It quantifies the degree of variation or dispersion of the possible outcomes.
In practical terms, variance helps us understand how "risky" or inconsistent our payoff might be compared to the expected value.
To calculate variance for the Payoff \(X\) in our lottery example, the formula used is:
\[ \sigma_X^2 = \sum ((\text{value} - \mu_X)^2 \times \text{probability}) \]
For this example:

• Subtract the expected value \(0.50\) from each possible payoff.
• Square these deviations: \( (0 - 0.5)^2 \) and \( (500 - 0.5)^2 \).
• Multiply each squared deviation by its probability and sum them: \(0.25 \times 0.999 + 249500.25 \times 0.001\).

Adding these gives a total variance of \( \sigma_X^2 = 249.750 \).
This number indicates the average squared deviation of each ticket's payoff from the mean, providing insight into the overall spread and risk involved in the lottery.

The standard deviation is the square root of the variance and provides a measure of the average distance of each probability outcome from the mean.
In simpler terms, it tells us how much we can expect a typical result to differ from the expected value.
In the Tri-State Pick 3 lottery, the standard deviation of the Payoff \(X\) is calculated as:
\[ \sigma_X = \sqrt{249.750} \approx 15.80 \]
The standard deviation of \(\\(15.80\) signifies that while the expected value of a ticket is only \(\\)0.50\), individual ticket outcomes vary significantly, typically by about \(\$15.80\).
This highlights the large fluctuations in winnings, caused by the lottery's win/lose structure.
When examining net winnings after accounting for the ticket cost (\(W = X - 1\)), the expected value shifts, but the standard deviation remains constant: \( \sigma_W = 15.80 \).
This unchanged standard deviation confirms that the spread or risk associated with purchasing a ticket remains the same even after considering the play cost.

Discrete probability distributions deal with outcomes that can be counted and have distinct values. Such distributions are characterized by a finite number of outcomes with specific probabilities associated with each.
In the case of the Tri-State Pick 3 lottery, the discrete nature is evident as the outcomes are simply winning \(\\(500\) or getting nothing (\(\\)0\)).
Understanding discrete probability helps in setting realistic expectations and effective decision-making for games of chance.
Each potential outcome in a discrete probability distribution is mutually exclusive, meaning only one of the outcomes can occur per event.

• For our lottery example: Winning \(\\(500\) occurs with a probability of 0.001.
• Winning nothing, \(\\)0\), happens with a probability of 0.999.

The probabilities of all possible outcomes in a discrete distribution always add up to one.
With each additional ticket purchase, players encounter fresh probabilities and possibilities of those specific outcomes, precisely captured through the notion of discrete probability.