- The expected value is the sum of each payout multiplied by its probability: \(E = \sum (x_i \cdot p_i)\).
- Calculate each component: \(100 \times 0.171 = 17.1\), \(4800 \times 0.029 = 139.2\), and \(180000 \times 0.0002 = 36\).
- Sum the components: \(17.1 + 139.2 + 36 = 192.3\).
- Round the total to the nearest dollar to get \(\boxed{\$192}\).
Explanation
- Analyze the problem and given data
To find the expected payout of the game, we first analyze the given data. We have three winning scenarios with specific payouts and their corresponding probabilities:
- Payout of \(\$100\) with a probability of \(0.171\).
- Payout of \(\$4,800\) with a probability of \(0.029\).
- Payout of \(\$180,000\) with a probability of \(0.0002\).
Additionally, there is a remaining probability associated with a payout of \(\$0\). Since the sum of all probabilities must equal \(1\), this fourth outcome accounts for the remaining probability. However, since the payout for this outcome is \(\$0\), it does not contribute to the final expected value calculation.
- Calculate the expected value components
The expected value \(E\) of a random variable is calculated by summing the products of each possible outcome and its associated probability. Mathematically, this is expressed as:
\[E = \sum (x_i \cdot p_i)\]
In our case, the calculation is:
\[E = (\$100 \cdot 0.171) + (\$4,800 \cdot 0.029) + (\$180,000 \cdot 0.0002) + (\$0 \cdot p_4)\]
Where \(p_4\) is the probability of the \(\$0\) payout. We calculate the terms one by one:
- \(100 \cdot 0.171 = 17.1\)
- \(4800 \cdot 0.029 = 139.2\)
- \(180000 \cdot 0.0002 = 36\)
- \(0 \cdot p_4 = 0\)
- Calculate the final expected value and round to the nearest dollar
Now, we sum the individual components calculated in the previous step:
\[E = 17.1 + 139.2 + 36 + 0 = 192.3\]
The total expected payout of the game is \(\$192.30\). The question asks for the answer to the nearest dollar. Rounding \(\$192.30\) to the nearest dollar gives \(\$192\).
Examples
In financial planning or insurance, expected value is a fundamental concept used to evaluate the long-term average outcome of random events. For example, if you are considering investing in a project with various potential returns (e.g., a \(5\%\) chance of gaining \(\$1,000\), a \(10\%\) chance of gaining \(\$500\), and an \(85\%\) chance of losing \(\$100\)), the expected return would be calculated as: \(E = (1000 \times 0.05) + (500 \times 0.10) + (-100 \times 0.85) = 50 + 50 - 85 = \$15\). This helps decision-makers determine if an investment is statistically profitable.