Questions: You buy a coffee maker at Best Buy. You are offered a warranty for 1 year that costs 15. If the coffee maker stops working within the warranty period, Best Buy will replace it. The cost of replacing the coffee maker is 80. The probability of product failure during the warranty period is 10%. Write the probability distribution table from the perspective of Best Buy.

Solution Steps

We consider two possible outcomes for Best Buy regarding the coffee maker warranty:

1. The coffee maker does not fail, resulting in a profit of \(-15\) (the cost of the warranty).
2. The coffee maker fails, resulting in a profit of \(65\) (the cost of the warranty minus the replacement cost).

The probabilities associated with these outcomes are:

• Probability of no failure: \(P(\text{no failure}) = 0.9\)
• Probability of failure: \(P(\text{failure}) = 0.1\)

The mean (expected value) of the profit can be calculated as follows:

\[ \text{Mean} = E(X) = (-15) \times 0.9 + 65 \times 0.1 = -7.0 \]

The variance of the profit is calculated using the formula:

\[ \text{Variance} = \sigma^2 = \sum (x_i - \mu)^2 P(x_i) \]

Substituting the values:

\[ \sigma^2 = (-15 - (-7.0))^2 \times 0.9 + (65 - (-7.0))^2 \times 0.1 \]

Calculating each term:

\[ = (-8)^2 \times 0.9 + (72)^2 \times 0.1 = 64 \times 0.9 + 5184 \times 0.1 = 57.6 + 518.4 = 576.0 \]

The standard deviation is the square root of the variance:

\[ \text{Standard Deviation} = \sigma = \sqrt{576.0} = 24.0 \]

Final Answer