Exercise 1.5: Drawing Cards

Wish result "Three aces"

From a deck of $32$ cards, including four aces, three cards are drawn in succession.

For subtask (1) it is assumed that after drawing a card it is put back into the deck, then the deck is reshuffled and the next card is drawn.

In contrast, for the other subtasks starting with (2) , you are supposed to assume that the three cards are drawn at once ("draw without putting back").

In the following, we denote by $A_i$ the event that the card drawn at time $i$ is an ace. Here $i \in \{ 1, 2, 3 \}$. The complementary event then states that some card other than an ace is drawn at time $i$ .

Hints:

The exercise belongs to the chapter Statistical dependence and independence.

The topic of this chapter is illustrated with examples in the (German language) learning video

Statistische Abhängigkeit und Unabhängigkeit $\Rightarrow$ "Statistical dependence and independence".

Questions

$p_1 \ = \ $

$p_2 \ = \ $

$p_3 \ = \ $

$p_4 \ = \ $

$p_5 \ = \ $

Solution

(1) For each card, the probability of an ace is exactly equal $(1/8)$:

$$p_{\rm 1} = {\rm Pr} (3 \hspace{0.1cm} {\rm Asse}) = {\rm Pr} (A_{\rm 1})\cdot {\rm Pr} (A_{\rm 2})\cdot {\rm Pr}(A_{\rm 3}) = \rm ({1}/{8})^3 \hspace{0.15cm}\underline {\approx 0.002}.$$

(2) Now, using the general multiplication theorem, we obtain:

$$p_{\rm 2} = {\rm Pr} (A_{\rm 1}\cap A_{\rm 2} \cap A_{\rm 3} ) = {\rm Pr} (A_{\rm 1}) \cdot {\rm Pr} (A_{\rm 2}\hspace{0.05cm}|\hspace{0.05cm}A_{\rm 1} ) \cdot {\rm Pr} \big[A_{\rm 3} \hspace{0.05cm}|\hspace{0.05cm}( A_{\rm 1}\cap A_{\rm 2} )\big].$$

The conditional probabilities are computable according to the classical definition. For this, one obtains $k/m$ (with $m$ cards, there are still $k$ aces in the deck):

$$p_{\rm 2} ={4}/{32}\cdot {3}/{31}\cdot{2}/{30} \hspace{0.15cm}\underline { \approx 0.0008}.$$

We can see: $p_2$ is smaller than $p_1$,, since now the second and third aces are less probable than before.

(3) Analogous to subtask (2) , we obtain here:

$$p_{\rm 3} = {\rm Pr}(\overline{A_{\rm 1}})\cdot {\rm Pr} (\overline{A_{\rm 2}} \hspace{0.05cm}|\hspace{0.05cm}\overline{A_{\rm 1}})\cdot {\rm Pr} (\overline{A_{\rm 3}}\hspace{0.05cm}|\hspace{0.05cm}(\overline{A_{\rm 1}} \cap \overline{A_{\rm 2}} )) = {28}/{32}\cdot{27}/{31}\cdot {26}/{30}\hspace{0.15cm}\underline {\approx 0.6605}.$$

(4) This probability can be expressed as the sum of three probabilities, since the associated events are disjoint:

$$p_{\rm 4} = {\rm Pr} (D_{\rm 1} \cup D_{\rm 2} \cup D_{\rm 3}) \rm \hspace{0.1cm}mit\hspace{-0.1cm}:$$

$$ {\rm Pr} (D_{\rm 1}) = {\rm Pr}( A_{\rm 1} \cap \overline{ A_{\rm 2}} \cap \overline{A_{\rm 3}}) = \rm \frac{4}{32}\cdot \frac{28}{31}\cdot \frac{27}{30}=\rm 0.1016,$$

$${\rm Pr} (D_{\rm 2}) = \rm Pr ( \overline{A_{\rm 1}} \cap A_{\rm 2} \cap \overline{A_{\rm 3}}) = \rm \frac{28}{32}\cdot \frac{4}{31}\cdot \frac{27}{30}=\rm 0.1016,$$

$${\rm Pr} (D_{\rm 3} \rm) = Pr ( \overline{\it A_{\rm 1}} \cap \overline{\it A_{\rm 2}} \cap A_{\rm 3}) = \rm \frac{28}{32}\cdot \frac{27}{31}\cdot \frac{4}{30}=\rm 0.1016.$$

These probabilities are all the same – why should it be any different?

If you draw exactly one ace from three cards, it is just as likely whether you draw it first, second, or third.

This gives $p_4 \; \underline{= 0.3048}$ for the sum.

(5) If we define the events $E_i :=$ "Exactly $i$ aces are drawn on three cards" with index $i \in \{ 0, 1, 2, 3 \}$,
then $E_0$, $E_1$, $E_2$ and $E_3$ describe a complete system. Therefore:

$$p_{\rm 5} = {\rm Pr}(E_2) = 1 - {\rm Pr}(E_0) -{\rm Pr}(E_1) - {\rm Pr}(E_3) = 1 - p_3 -p_4 - p_2 \hspace{0.15cm}\underline {= \rm 0.0339}.$$