Probability Playground#
A standard Deck of Playing Cards makes a great sandbox for a probability course. Most probability ideas can be quickly illustrated using a deck of cards and random draws.The standard deck has ready-made subsets and partitions which help illustrate important ideas. For those unfamiliar with a standard deck of playing cards, there are the thirteen values (or ranks) and four suits: \(\spadesuit,\textcolor{red}{\heartsuit}, \clubsuit,\textcolor{red}{\diamondsuit}\).
Graphics: Deck of Cards
Whenever we refer to “drawing a card,” we mean that a standard deck of 52 playing cards has been well-shuffled and that the draw is perfectly random. Each card has an equal probability of being drawn. We never include jokers or wild cards.The entire deck of cards is shown in the graphic below:
Half the cards are red, half black. Each suit subset has 13 cards in it. Each value subset has four cards. For example, the set \(J=\{J\spadesuit,J\heartsuit, J\clubsuit,J\diamondsuit\}\)
Example 4
Suppose we draw a single card at random from a well-shuffled standard deck of 52 cards. What is the probability of drawing a heart? Of \emph{not} drawing a heart?
Solution. Let the probability space \(S\) be comprised of all possible outcomes when drawing one card at random from a standard deck of playing cards. Let \(H\) be the event set of drawing a heart (\(\heartsuit\)). Then \(|H|=13\) because there are 13 possible \(\heartsuit\) outcomes in \(S\). Thus:
The probability of “not drawing a heart” inquires about the probability of the set complement of the event set \(H\) and is denoted \(\bar H\). The probability of the complement of an event set is given by:
Because \(|\bar H|=|S|-|H|\), we have:
To conclude the example from above:
In working with a deck of cards, certain common sets can helpful:
F: Face cards (all Kings, Queens and Jacks)
E: Even cards (2’s, 4’s, 6’s, 8’s and T’s)
O: Odd cards (3’s, 5’s, 7’s, 9’s)
Suits as in Hearts (H), Spades (S), Clubs (C) and Diamonds (D)
R: Red cards _ B: Black cards