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The concept of expected value is a cornerstone in probability and statistics. It is essentially the "average" outcome you can anticipate from a random event over many trials. In the context of the Keno game, we use expected value to determine the average net gain or loss per game played.

To calculate the expected value, we multiply each outcome (like winning or losing) by its respective probability. Then, we sum these products to find the overall expected outcome. For Keno, our random variable, X, can take on two values—that is, winning \(\\(2\) or losing \(\\)1\).

The expected value formula for this game looks like this:

• Outcome 1: Win \(\\(2\) (net gain), with probability \(0.25\)
• Outcome 2: Lose \(\\)1\) (net loss), with probability \(0.75\)

So, the calculation is:\[E(X) = (2 \times 0.25) + (-1 \times 0.75) = 0.5 - 0.75 = -0.25\]This negative result indicates an average loss of \(\$0.25\) per game over time, highlighting the risk involved in playing.

Random variables are a fundamental part of probability theory. They are variables that quantify the outcomes of a probabilistic event, and they can take on different values based on the event's result. In the Keno game, we define the random variable \(X\) to represent the net gain from a single play.

In this scenario, \(X\) is a discrete random variable, meaning it takes on countable values. Specifically, \(X\) can be either \(2\) if you win (since you win \(\\(3\) but pay \(\\)1\) to play), or \(-1\) if you lose (since you lose the \(\$1\) spent to bet).

Random variables help in setting up and solving probability-related problems. By knowing the values \(X\) can take and their respective probabilities, we can perform calculations such as finding the expected value, which gives us insights into the long-term outcomes of the game.

Probability calculates the likelihood of an event occurring. When dealing with games like Keno, understanding probability helps us assess the odds and make informed decisions.

For Keno, calculating the probability of winning is straightforward since numbers are drawn randomly. The probability of winning—of having your chosen number appear among the 20 drawn out of 80—is \(0.25\), or simply \(\frac{20}{80}\). Consequently, the probability of losing (not having your number drawn) is \(0.75\), calculated as \(1 - 0.25\).

These probabilities allow us to calculate the expected value. Moreover, by comprehending these calculations, you can apply similar reasoning to other probabilistic scenarios, making the math behind games of chance less mysterious and more manageable.