Question

Suppose that from a standard deck, you draw $3$ cards without replacement. What is the expected number of spades that you will draw?

Answer 1

Find the expected number of spades that you will draw from a standard deck.

The total number of cards in a deck is $52$.

Given that $3$ cards are to be drawn without replacement.

Total spade cards in a deck are $13$.

Apply the formula $\mathrm{P}\left(\mathrm{Spade}\mathrm{on}\mathrm{first}\mathrm{draw}\right)=\frac{\mathrm{Number}\mathrm{of}\mathrm{spade}\mathrm{cards}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{cards}}$;

$$\begin{gathered} \mathrm{P}\left(\mathrm{Spade}\mathrm{on}\mathrm{first}\mathrm{draw}\right)=\frac{13}{52} \\ \mathrm{P}\left(\mathrm{Spade}\mathrm{on}\mathrm{first}\mathrm{draw}\right)=\frac{1}{4} \end{gathered}$$

Since, the card is not replaced after the first draw the remaining spade cards are $12$ and the total number of remaining cards are $51$.

Hence, $\mathrm{P}\left(\mathrm{Spade}\mathrm{on}\mathrm{second}\mathrm{draw}\right)=\frac{12}{51}$.

Now in the third draw $11$ spade cards are left and the total number of remaining cards are $50$.

$\mathrm{P}\left(\mathrm{Spade}\mathrm{on}\mathrm{third}\mathrm{draw}\right)=\frac{11}{50}$.

Now apply the formula $\mathrm{P}\left(3\mathrm{spade}\mathrm{cards}\mathrm{without}\mathrm{replacement}\right)=\mathrm{P}\left(\mathrm{Spade}\mathrm{on}\mathrm{first}\mathrm{draw}\right)\times \mathrm{P}\left(\mathrm{Spade}\mathrm{on}\mathrm{second}\mathrm{draw}\right)\times \mathrm{P}\left(\mathrm{Spade}\mathrm{on}\mathrm{third}\mathrm{draw}\right)$,

Substitute the known values;

$\begin{array}{rcl}\mathrm{P}\left(3\mathrm{spade}\mathrm{cards}\mathrm{without}\mathrm{replacement}\right)& =& \frac{1}{4}\times \frac{12}{51}\times \frac{11}{50}\\ & =& \frac{33}{2550}\\ & =& 0.0129\end{array}$

Hence, the expected number of spades that will be drawn is $0.0129$.