To calculate the expected net winnings for a $25 bet on black, we need to apply the concept of expected value, which considers both the probabilities of outcomes and the associated monetary gains or losses.

The probability of a black outcome is given as $P(\text{black}) = \frac{18}{38}$.

1. Possible outcomes:

• If the ball lands on black, the player wins $25 (the amount bet) in addition to the original bet, for a total of $50.
• If the ball lands on red, the player loses the $25 bet, so the outcome is -$25.
• If the ball lands on green, the player also loses the $25 bet, for a total of -$25.

2. Expected value formula:

$E(\text{Winnings}) = (P(\text{black}) \times \text{Winnings on black}) + (P(\text{red}) \times \text{Winnings on red}) + (P(\text{green}) \times \text{Winnings on green})$

Substitute the values:

• $P(\text{black}) = \frac{18}{38}$, Winnings on black = $50
• $P(\text{red}) = \frac{18}{38}$, Winnings on red = -$25
• $P(\text{green}) = \frac{2}{38}$, Winnings on green = -$25

$E(\text{Winnings}) = \left(\frac{18}{38} \times 50\right) + \left(\frac{18}{38} \times (-25)\right) + \left(\frac{2}{38} \times (-25)\right)$

Let's calculate it:

$E(\text{Winnings}) = \left( \frac{18}{38} \times 50 \right) + \left( \frac{18}{38} \times (-25) \right) + \left( \frac{2}{38} \times (-25) \right)$

$E(\text{Winnings}) = \left( \frac{18 \times 50}{38} \right) + \left( \frac{18 \times (-25)}{38} \right) + \left( \frac{2 \times (-25)}{38} \right)$

$E(\text{Winnings}) = \frac{900}{38} - \frac{450}{38} - \frac{50}{38}$

$E(\text{Winnings}) = \frac{900 - 450 - 50}{38} = \frac{400}{38}$

$E(\text{Winnings}) \approx 10.526$

Interpretation:

The expected net winnings for this single bet of $25 on black is approximately $10.526.

This means that, on average, for each bet placed on black in the long run, a player would expect to win about $10.53 per bet.

Correct Answer:

B. If the $25 bet on black was placed a very large number of times, the average winnings would be $10.526 per bet.

Let me know if you'd like further clarification or if you have more questions!

Related Questions:

1. How would the expected value change if the probability of landing on red were different?
2. What is the expected value of betting on red instead of black in this game?
3. How does the probability of winning affect the expected value for any bet?
4. What would the expected value be if the bet was placed on green instead of black or red?
5. How does increasing the bet amount (e.g., $50 instead of $25) affect the expected winnings?

Tip: When calculating expected value, always consider both the probability of each outcome and its respective payoff to get a complete picture of the long-term average result.