To find the expected value of the winnings from the given payout probability distribution, we can use the following steps:
1. Identify the Payouts and Probabilities:
[tex]\[
\begin{array}{c|ccccc}
\text { Payout }(\$) & 0 & 5 & 8 & 10 & 15 \\
\hline \text { Probability } & 0.50 & 0.20 & 0.15 & 0.10 & 0.05 \\
\end{array}
\][/tex]
2. Multiply Each Payout by Its Corresponding Probability:
- For the payout of \[tex]$0 with probability 0.50:
\[
0 \times 0.50 = 0
\]
- For the payout of \$[/tex]5 with probability 0.20:
[tex]\[
5 \times 0.20 = 1
\][/tex]
- For the payout of \[tex]$8 with probability 0.15:
\[
8 \times 0.15 = 1.2
\]
- For the payout of \$[/tex]10 with probability 0.10:
[tex]\[
10 \times 0.10 = 1
\][/tex]
- For the payout of \$15 with probability 0.05:
[tex]\[
15 \times 0.05 = 0.75
\][/tex]
3. Sum All the Products to Get the Expected Value:
[tex]\[
0 + 1 + 1.2 + 1 + 0.75 = 3.95
\][/tex]
4. Round the Expected Value to the Nearest Hundredth:
The sum we obtained is already to two decimal places, so no further rounding is needed.
Therefore, the expected value of the winnings from this game is:
[tex]\[
\boxed{3.95}
\][/tex]
