In a game of cards you win $\$ 1$ if you draw a heart, $\$ 5$ if you draw an ace (including the ace of hearts), $\$ 10$ if you draw the king of spades and nothing for any other card you draw. Use the table below and calculate your expected winning. \begin{tabular}{l|c|c|c} Event & $X$ & $P(X)$ & $X P(X)$ \\ \hline Heart (not ace) & 1 & $\frac{12}{52}$ & \\ Ace & 5 & $\frac{4}{52}$ & \\ King of spades & 10 & $\frac{1}{52}$ & \\ All else & 0 & $\frac{35}{52}$ & \end{tabular} 0.81 1 4.923 0.308

Step 1 ：Given the payouts and probabilities for each event in the game of cards, we can calculate the expected winning by multiplying the payout for each event by its probability and then summing these products.

Step 2 ：The payouts are: \$1 for a heart (not an ace), \$5 for an ace, \$10 for the king of spades, and \$0 for all other cards.

Step 3 ：The probabilities for each event are: \(\frac{12}{52}\) for a heart (not an ace), \(\frac{4}{52}\) for an ace, \(\frac{1}{52}\) for the king of spades, and \(\frac{35}{52}\) for all other cards.

Step 4 ：We calculate the product of the payout and probability for each event: \(1 \times \frac{12}{52} = 0.23076923076923078\), \(5 \times \frac{4}{52} = 0.38461538461538464\), \(10 \times \frac{1}{52} = 0.19230769230769232\), and \(0 \times \frac{35}{52} = 0.0\).

Step 5 ：We sum these products to get the expected winning: \(0.23076923076923078 + 0.38461538461538464 + 0.19230769230769232 + 0.0 = 0.8076923076923077\).

Step 6 ：The expected winning in the game of cards is approximately \$0.81. Therefore, the final answer is \(\boxed{0.81}\).