To find the expected value of the winnings from the given game, we will use the concept of expected value in probability theory. The expected value (or mean) of a discrete random variable is calculated by multiplying each possible outcome by its probability and then summing all these products.
Given the payout probability distribution:
[tex]\[
\begin{array}{c|ccccc}
\text {Payout (\$)} & 0 & 2 & 4 & 6 & 8 \\
\hline
\text {Probability} & 0.36 & 0.06 & 0.33 & 0.08 & 0.17 \\
\end{array}
\][/tex]
We calculate the expected value ([tex]\(E\)[/tex]) as follows:
[tex]\[
E = (0 \times 0.36) + (2 \times 0.06) + (4 \times 0.33) + (6 \times 0.08) + (8 \times 0.17)
\][/tex]
Step-by-step, we compute each term:
1. [tex]\(0 \times 0.36 = 0\)[/tex]
2. [tex]\(2 \times 0.06 = 0.12\)[/tex]
3. [tex]\(4 \times 0.33 = 1.32\)[/tex]
4. [tex]\(6 \times 0.08 = 0.48\)[/tex]
5. [tex]\(8 \times 0.17 = 1.36\)[/tex]
Next, sum all these products:
[tex]\[
E = 0 + 0.12 + 1.32 + 0.48 + 1.36
\][/tex]
[tex]\[
E = 3.2800000000000002
\][/tex]
To provide the final answer, we round the expected value to the nearest hundredth:
[tex]\[
E \approx 3.28
\][/tex]
Therefore, the expected value of the winnings from the game, rounded to the nearest hundredth, is [tex]\(3.28\)[/tex] dollars.
