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The expected value is a crucial concept when dealing with probability distributions, as it gives insight into what we can "expect" on average from a probability scenario. In the context of our roulette wheel problem, we compute the expected value, denoted as \(E(X)\), to determine an average outcome for the bettor placing a \(1 bet on red.When calculating the expected value, we use the formula:\[E(X) = \sum (x_i \times P(x_i))\]where \(x_i\) are the possible outcomes and \(P(x_i)\) are their respective probabilities. For our problem:

• The bettor gains \)1 with a probability of \(\frac{18}{38}\)
• The bettor loses \(1 with a probability of \(\frac{20}{38}\)

Plugging in the values, we have:\[ E(X) = (-1) \times \frac{20}{38} + 1 \times \frac{18}{38} = \frac{-2}{38} = -\frac{1}{19} \approx -0.0526 \]This negative expected value means that, on average, the bettor will lose approximately \\)0.0526 per bet. So the expected value highlights the concept of the house edge, in which the casino typically gains in the long run.

Standard deviation is a measure of the spread or dispersion of a set of values around the mean in a probability distribution. It tells us how much on average, outcomes differ from the expected value, \(E(X)\), in our betting scenario.To find the standard deviation, \(\sigma\), we first compute the variance, \(\sigma^2\), using this formula:\[ \sigma^2 = \sum ((x_i - E(X))^2 \times P(x_i)) \]Substitute the values:

• The deviation of \(-1\) from \(-\frac{1}{19}\) squared, times its probability \(\frac{20}{38}\)
• The deviation of \(1\) from \(-\frac{1}{19}\) squared, times its probability \(\frac{18}{38}\)

Then compute the variance:\[ \sigma^2 = \frac{360}{361} \times \frac{20}{38} + \frac{400}{361} \times \frac{18}{38} \]Finally, the standard deviation is the square root of the variance:\[ \sigma = \sqrt{\sigma^2} \]The value of standard deviation shows us how much the actual payout is likely to differ with each bet on red. A larger standard deviation implies greater variability, which translates to a riskier bet.

Variance is fundamental in understanding the variability of outcomes in a probability distribution. Like standard deviation, variance explains how spread out your results can be from the mean (expected value), but it does so in the form of squared units.For our roulette example, variance helps us understand the volatility or risk associated with betting on red. To calculate variance, we employ the following formula:\[ \sigma^2 = \sum ((x_i - E(X))^2 \times P(x_i)) \]Stepping through our calculations:

• Compute the square of the deviation of each outcome (\(-1\) and \(1\)) from the expected value \(-\frac{1}{19}\)
• Multiply each squared deviation by its corresponding probability (\(\frac{20}{38}\) and \(\frac{18}{38}\))

This results in the exact variance value, which serves as a precursor to finding the standard deviation. Variance also allows financial analysts and bettors to evaluate the riskiness or stability of a venture over time. By understanding variance, one can anticipate how much a betting strategy could potentially deviation from its expected gain or loss.