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A strange deck of cards
In a non-standard deck of cards there are
- 20 blue cards (numbered 1 through 20),
- 20 green cards (numbered 1 through 20), and
20 red cards (numbered 1 through 20)
Four cards are dealt without replacement from this deck.
- What is the probability that exactly two of the four cards dealt are blue?
- Given that at least one of the first two cards dealt is blue, what is the probability that exactly three of the four cards dealt are blue?
- What is the probability that at least two of the four cards dealt have the same numeric value (1 through 20)?
Which deck is rigged ?
Two decks of cards are sitting on a table. One deck is a standard deck of 52 cards. The other deck (called the rigged deck) also has 52 cards but has had 4 of the 13 Harts replaced by Diamonds. (Recall that a standard deck has 4 suits: Diamonds, Harts, Spades, and Clubs. normal there are 13 of each suit.)
- What is the probability one chooses 4 cards from the rigged deck and gets exactly 2 diamonds and no hearts?
- What is the probability one chooses 4 cards from the standard deck and gets exactly 2 diamonds and no hearts?
- You randomly chose one of the decks and draw 4 cards. You obtain exactly 2 diamonds and no hearts.
- What is the probability you chose the cards from the rigged deck?
- What is the probability you chose the cards from the standard deck?
- If you had to guess which deck was used, which would you guess? The standard or the rigged ?
Independence of two hearts ?
Consider a deck of 52 cards. Let \(A\) be the event that the first card is a heart. Let \(B\) be the event that the 51st card is a heart.
What is \(\mathbf{P}(A)\) ? What is \(\mathbf{P}(B)\) ? Are \(A\) and \(B\) independent ?
Cards again
Given a well shuffled standard deck of 52 cards, what is the probability of what of the following events. (Think before you jump.)
- The 1st card is an ace.
- The 15th card is an ace.
- The 9th card is a diamond.
- The last 5 cards are hearts.
- The 17th card is the ace of diamonds and the 14 is the King of spades
- The 5th card is a diamond given that the 50th card is a diamond.
Time to play some bridge!
A hand in bridge consists of thirteen cards dealt out from a well shuffled deck.
(a) What is the probability that the bridge hand contains exactly 5 hearts?
(b) What is the probability that the bridge hand contains exactly 5 hearts and 5 spades?
(c) What is the probability that the hand contains exactly 5 cards from at least one suit?
[Author Mark Huber. Licensed under Creative Commons.]
Poker Hands: counting
Assume that each of Poker hands are equally likely. The total number of hands is
\[\begin{pmatrix} 52 \\5\end{pmatrix}\]
Find the probability of being dealt each of the following:
- a straight flush ( all cards of the same suit and in order)
- a regular straight (but not a flush)
- two of a kind
- four of a kind
- two pairs (but not four of a kind)
- a full house (a pair and three of a kind)
In all cases, we mean exactly the hand stated. For example, four of a kind does not count as 2 pairs and a full house does not count as a pair or three of a kind.
Card hands: court cards
In a hand of 13 cards drawn randomly from a pack of 53, find the chance of:
- no court cards (J,Q,K,A);
- at least one ace but no other court cards;
- at most one kind of court card.
[Pitman p. 128, # 6]
