We define the random variable $X$ as the net winnings from a single bet in roulette.

Step 1: Define the Probability Distribution

There are 38 slots on the roulette wheel, and we consider the following two possible outcomes:

1. Winning Outcome:

   • Probability of winning: $P(\text{win}) = \frac{1}{38}$
   • Winnings: $35

2. Losing Outcome:

   • Probability of losing: $P(\text{loss}) = \frac{37}{38}$
   • Loss: $-1

Step 2: Compute the Mean (Expected Value)

The expected value $E(X)$ (mean) is given by:

$E(X) = (35 \times P(\text{win})) + (-1 \times P(\text{loss}))$

$E(X) = \left( 35 \times \frac{1}{38} \right) + \left( -1 \times \frac{37}{38} \right)$

$E(X) = \frac{35}{38} - \frac{37}{38} = \frac{35 - 37}{38} = \frac{-2}{38} = -0.0526$

So, the expected value (mean) is $-0.05$ dollars (rounded to the nearest cent), meaning a player loses, on average, 5.26 cents per bet.

Step 3: Compute the Variance and Standard Deviation

The variance is given by:

$\sigma^2 = E(X^2) - (E(X))^2$

First, we compute $E(X^2)$:

$E(X^2) = (35^2 \times P(\text{win})) + ((-1)^2 \times P(\text{loss}))$

$E(X^2) = \left( 1225 \times \frac{1}{38} \right) + \left( 1 \times \frac{37}{38} \right)$

$E(X^2) = \frac{1225}{38} + \frac{37}{38} = \frac{1225 + 37}{38} = \frac{1262}{38} \approx 33.21$

Now, compute the variance:

$\sigma^2 = 33.21 - (-0.0526)^2$

$\sigma^2 = 33.21 - 0.00277 \approx 33.21$

Finally, the standard deviation $\sigma$ is:

$\sigma = \sqrt{33.21} \approx 5.76$

Final Answers:

• Mean (Expected Value): $-0.05$ dollars (or $-5.26$ cents)
• Standard Deviation: $5.76

Would you like a breakdown of how this affects long-term play? Let me know if you have any questions!

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Related Questions:

1. How does the house edge in roulette compare to other casino games?
2. If a player bets 100 times, what is their expected total loss?
3. How would the expected value change if the payout for winning were $36 instead of $35?
4. What is the probability of winning at least once in 10 bets?
5. How does the standard deviation impact the short-term variance in winnings?

Tip:

Always consider expected value when playing games of chance—it tells you how much you will win or lose on average per bet!