Questions: Step 1 of 2: What is the expected value of your bet? Round your answer to two decimal places. Losses must be expressed as negative values.

Solution Steps

To find the expected value (mean) of the bet, we use the formula:

\[ \text{Mean} = \sum (x_i \cdot p_i) \]

where \( x_i \) are the possible outcomes and \( p_i \) are their corresponding probabilities. For our bet:

\[ \text{Mean} = 100 \times 0.4 + (-50) \times 0.6 = 40 - 30 = 10.0 \]

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \sum ((x_i - \text{Mean})^2 \cdot p_i) \]

Substituting the values:

\[ \sigma^2 = (100 - 10.0)^2 \times 0.4 + (-50 - 10.0)^2 \times 0.6 \]

Calculating each term:

\[ = (90)^2 \times 0.4 + (-60)^2 \times 0.6 = 8100 \times 0.4 + 3600 \times 0.6 = 3240 + 2160 = 5400.0 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{5400.0} \approx 73.48 \]

Final Answer