Sam's Dilemma: Choosing Paint Colors for a House

Sam, the builder, is currently facing a crucial decision in choosing paint colors and finishes for a house he is working on. The homeowner's requirement for the paint and trim colors to match adds an additional layer of complexity to Sam's task. If the colors do not match, Sam will have to redo the work, leading to significant expenses.

Sam's options include 6 paint colors: red, green, white, black, purple, and yellow, and 3 finishing options: red, green, and white. The homeowner will be satisfied if Sam selects the same color for both the paint and trim, earning Sam a reward of $1000. However, if Sam opts for different colors, he will incur a penalty of $100.

The probability of choosing a matching color for both the paint and trim is 3 out of 6, or 50%. To calculate Sam's expected value, we utilize the formula: (Probability of success × Winning amount) + [(1 - Probability of success) × Losing amount].

Therefore, the expected value is calculated as follows: (0.5 × $1000) + (0.5 × -$100) = $500 + (-$50) = $450. This indicates that if Sam makes random choices, he can expect an average net gain of $450 when considering the probabilities of winning and losing.

Given the 50% chance of selecting matching colors, it is advisable for Sam to make careful and thoughtful choices to maximize his expected value and avoid unnecessary costs. By choosing wisely, Sam can enhance his chances of achieving a positive outcome and securing a reward of $1000 without incurring any additional expenses.