Questions: QUESTION 2 · 1 POINT A casino features a game in which a weighted coin is tossed several times. The table shows the probability of each payout amount. To the nearest dollar, what is expected payout of the game? Payout Amount Probability 200 0.126 3800 0.03 190000 0.0002

Solution Steps

To find the expected payout of the game, we compute the mean using the formula:

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

where \( x_i \) represents the payout amounts and \( p_i \) represents their corresponding probabilities. Substituting the values:

\[ \text{Mean} = 200 \times 0.126 + 3800 \times 0.03 + 190000 \times 0.0002 \]

Calculating each term:

\[ 200 \times 0.126 = 25.2 \] \[ 3800 \times 0.03 = 114 \] \[ 190000 \times 0.0002 = 38 \]

Adding these results together:

\[ \text{Mean} = 25.2 + 114 + 38 = 177.2 \]

Rounding to the nearest dollar gives us:

\[ \text{Expected payout} = 177 \]

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

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

Substituting the values:

\[ \sigma^2 = (200 - 177)^2 \times 0.126 + (3800 - 177)^2 \times 0.03 + (190000 - 177)^2 \times 0.0002 \]

Calculating each term:

\[ (200 - 177)^2 \times 0.126 = 23^2 \times 0.126 = 529 \times 0.126 = 66.834 \] \[ (3800 - 177)^2 \times 0.03 = 3623^2 \times 0.03 = 13120229 \times 0.03 = 393606.87 \] \[ (190000 - 177)^2 \times 0.0002 = 189823^2 \times 0.0002 = 36000000000 \times 0.0002 = 7200000 \]

Adding these results together:

\[ \sigma^2 = 66.834 + 393606.87 + 7200000 = 7600345.0 \]

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

\[ \sigma = \sqrt{\sigma^2} = \sqrt{7600345.0} \approx 2757.0 \]

Final Answer