Guessing Cards: Aces and Kings
Phil is told there are 3 Aces and 3 Kings in a pile of cards. In each turn, a card is drawn without replacement, and Phil earns $1 if he correctly guesses the drawn card. Using an optimal strategy, he plays 6 turns. What is his expected earning?
Answer
Let \(X_i\) be the indicator that Phil’s guess on turn \(i\) is correct. By linearity of expectation, his total expected earning is
\[
\mathbb{E}\left[\sum_{i=1}^6 X_i\right] = \sum_{i=1}^6 \mathbb{E}[X_i].
\]
Using the optimal strategy—guessing the card type (Ace or King) that is more likely to appear next—we obtain the following expected probabilities for each turn:
\[
\begin{alignedat}{9}
\mathbb{E}[X_1] &{}={}& \frac{1}{2},\quad
&\mathbb{E}[X_2] &{}={}& \frac{3}{5},\quad
&\mathbb{E}[X_3] &{}={}& \frac{3}{5},\quad
&\mathbb{E}[X_4] &{}={}& \frac{7}{10},\quad
&\mathbb{E}[X_5] &{}={}& \frac{7}{10},\quad
&\mathbb{E}[X_6] &{}={}& 1.
\end{alignedat}
\]
Summing these, we find that
\[
\mathbb{E}\left[\sum_{i=1}^6 X_i\right] = 0.5 + 0.6 + 0.6 + 0.7 + 0.7 + 1 = 4.1.
\]
Thus, Phil’s expected earning is \(\boxed{\$4.10}\).