To find the expected value of the winnings from the given game, we need to use the concept of expected value in probability theory. The expected value (EV) of a random variable is a measure of the center of its probability distribution and is calculated by summing the products of each outcome and its corresponding probability.
Here is the step-by-step solution:
1. Identify the payouts and probabilities:
[tex]\[
\begin{aligned}
&\text{Payouts}:\quad &&0\, \$,\, 2\, \$,\, 5\, \$,\, 6\, \$,\, 10\, \$ \\
&\text{Probabilities}:\quad &&0.57,\, 0.21,\, 0.04,\, 0.04,\, 0.14
\end{aligned}
\][/tex]
2. Multiply each payout by its probability to find the contribution of each outcome to the expected value:
[tex]\[
\begin{aligned}
&0 \times 0.57 = 0 \\
&2 \times 0.21 = 0.42 \\
&5 \times 0.04 = 0.20 \\
&6 \times 0.04 = 0.24 \\
&10 \times 0.14 = 1.40 \\
\end{aligned}
\][/tex]
3. Sum these contributions to get the total expected value:
[tex]\[
\begin{aligned}
EV &= 0 + 0.42 + 0.20 + 0.24 + 1.40 \\
&= 2.26
\end{aligned}
\][/tex]
So, the expected value of the winnings from the game is [tex]\( 2.26 \)[/tex] dollars.
4. Round to the nearest hundredth where necessary:
In this case, [tex]\( 2.26 \)[/tex] is already rounded to the nearest hundredth.
Therefore, the expected value of the winnings is:
[tex]\[
\boxed{2.26}
\][/tex]
