To find the expected value of the winnings from the game, you need to follow the steps for calculating the expected value of a discrete random variable. The expected value (E) is calculated by multiplying each possible payout by its corresponding probability and then summing up all these products.
Let's denote the payouts as [tex]\( X_i \)[/tex] and the probabilities as [tex]\( P(X_i) \)[/tex].
Here are the payouts and their corresponding probabilities:
[tex]\[
\begin{array}{c|c|c|c|c|c}
\text{Payout (\$)} & 3 & 4 & 5 & 6 & 7 \\ \hline
\text{Probability} & 0.01 & 0.04 & 0.10 & 0.20 & 0.65 \\
\end{array}
\][/tex]
The expected value [tex]\( E(X) \)[/tex] is computed as:
[tex]\[
E(X) = \sum_{i} X_i \cdot P(X_i)
\][/tex]
Calculating each term individually:
1. For payout [tex]\( X_1 = 3 \)[/tex]:
[tex]\[
3 \times 0.01 = 0.03
\][/tex]
2. For payout [tex]\( X_2 = 4 \)[/tex]:
[tex]\[
4 \times 0.04 = 0.16
\][/tex]
3. For payout [tex]\( X_3 = 5 \)[/tex]:
[tex]\[
5 \times 0.10 = 0.50
\][/tex]
4. For payout [tex]\( X_4 = 6 \)[/tex]:
[tex]\[
6 \times 0.20 = 1.20
\][/tex]
5. For payout [tex]\( X_5 = 7 \)[/tex]:
[tex]\[
7 \times 0.65 = 4.55
\][/tex]
Now, sum these values together to find the expected value:
[tex]\[
E(X) = 0.03 + 0.16 + 0.50 + 1.20 + 4.55 = 6.44
\][/tex]
Therefore, the expected value of the winnings from the game is:
[tex]\[
\boxed{6.44}
\][/tex]
