[Math] Drawing a card from a deck

probability

A single card is drawn from a standard 52-deck of cards with four suits: hearts, clubs, diamonds, and spades; there are 13 cards per suit. If each suit has three face cards, how many ways could the drawn card be either a club of any kind or anything else besides a face card?

Best Answer

I have a feeling your teacher/professor intended for you to learn the following (we first introduce some notation):

For two sets $A, B$ in a common universe $U$, define their union as

$$ A \cup B = \{x \in U : x \in A \text { or } x \in B\}. $$

Define their intersection as:

$$ A \cap B = \{x \in U : x \in A \text{ and } x \in B\}. $$

It is important not to get too wrapped in the English here. Being in $A \cup B$ simply means being in one of the two sets (or possibly both). Being in $A \cap B$ simply means being in set $A$ and being in set $B$ at the same time.

Finally, a finite set $A$ with $k$ elements ($k$ things in the set) has cardinality $k$, and this is written as $|A| = k$, or $\#A = k$ or even sometimes, $n(A) = k$.

Therefore:

$$ \{1,3,5\} \cup \{1,2,3\} = \{1,2,3,5\}, $$

while

$$ \{1 ,3 , 5 \} \cap \{ 1 , 2, 3 \} = \{ 1, 3\}. $$

Also, $$ | \{ 1,3,5\}| = 3, $$

while

$$ | \{ 1, 3\} | = 2. $$

Now, what your teacher probably wanted you to learn was the following "rule":

$$ |A \cup B| = |A| + |B| - |A \cap B|. $$

This is easy to see it is true, since to count the number of elements that are in either $A$ or $B$ (or possibly both $A$ and $B$), you count the number of elements in $A$, add to it the number of elements in $B$, and then subtract the stuff you double counted, which is precisely the elements in $A \cap B$.

Therefore, if you want to find the number of sides of a die that are (say) even or prime, you count the number of sides which are even (there are $3$ such sides - namely $2$ , $4$, $6$) add to it the number of sides which are prime (again, there are $3$ such sides - namely $2$, $3$, $5$), and then subtract the sides which we double counted (we counted the side with the number $2$ twice).

Therefore, there are $3 + 3 - 1 = 5$ sides of a die which are even or prime.

Now, you can take this strategy and count the number of cards in a deck which are either non face cards or clubs.

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