To find the expected value of the winnings from the game, follow these steps:
1. Identify the payouts and their respective probabilities:
[tex]\[
\begin{array}{cc}
\text{Payout (\$)} & \text{Probability} \\
-1 & 0.70 \\
1 & 0.15 \\
3 & 0.10 \\
5 & 0.04 \\
7 & 0.01 \\
\end{array}
\][/tex]
2. Multiply each payout by its corresponding probability:
[tex]\[
\begin{array}{ccc}
\text{Payout} & \times & \text{Probability} \\
-1 & \times & 0.70 = -0.70 \\
1 & \times & 0.15 = 0.15 \\
3 & \times & 0.10 = 0.30 \\
5 & \times & 0.04 = 0.20 \\
7 & \times & 0.01 = 0.07 \\
\end{array}
\][/tex]
3. Sum these products to obtain the expected value:
[tex]\[
\text{Expected Value} = (-0.70) + 0.15 + 0.30 + 0.20 + 0.07
\][/tex]
4. Calculate the sum:
[tex]\[
\text{Expected Value} = -0.70 + 0.15 + 0.30 + 0.20 + 0.07 = 0.02
\][/tex]
Hence, the expected value of the winnings from the game is:
[tex]\[
\boxed{0.02}
\][/tex]
