Questions: Find the expected value of the winnings from a game that has the following payout probability distribution: Payout () 0 2 5 10 25 Probability 0.61 0.24 0.10 0.04 0.01 Expected Value =

Solution Steps

To find the expected value of the winnings from the game, we need to multiply each payout by its corresponding probability and then sum these products.

We are given the payouts and their corresponding probabilities: \[ \begin{array}{c|ccccc} \text{Payout (\$)} & 0 & 2 & 5 & 10 & 25 \\ \hline \text{Probability} & 0.61 & 0.24 & 0.10 & 0.04 & 0.01 \\ \end{array} \]

The expected value \(E\) of the winnings is calculated using the formula: \[ E = \sum_{i=1}^{n} (x_i \cdot p_i) \] where \(x_i\) are the payouts and \(p_i\) are the corresponding probabilities.

Substituting the given values: \[ E = (0 \cdot 0.61) + (2 \cdot 0.24) + (5 \cdot 0.10) + (10 \cdot 0.04) + (25 \cdot 0.01) \]

\[ \begin{align_} 0 \cdot 0.61 &= 0 \\ 2 \cdot 0.24 &= 0.48 \\ 5 \cdot 0.10 &= 0.50 \\ 10 \cdot 0.04 &= 0.40 \\ 25 \cdot 0.01 &= 0.25 \\ \end{align_} \]

\[ E = 0 + 0.48 + 0.50 + 0.40 + 0.25 = 1.63 \]

Final Answer