# Category Archives: Cards

## 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.

1. What is the probability that exactly two of the four cards dealt are blue?
2. 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?
3. 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.)

1. What is the probability one chooses 4 cards from the rigged deck and gets exactly 2 diamonds and no hearts?
2. What is the probability one chooses 4 cards from the standard deck and gets exactly 2 diamonds and no hearts?
3. You randomly chose one of the decks and draw 4 cards. You obtain exactly 2 diamonds and no hearts.
1. What is the probability you chose the cards from the rigged deck?
2. What is the probability you chose the cards from the standard deck?
3. 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.)

1. The 1st card is an ace.
2. The 15th card is an ace.
3. The 9th card is a diamond.
4. The last 5 cards are hearts.
5. The 17th card is the ace of diamonds and the 14 is the King of spades
6. 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:

1. a straight flush ( all cards of the same suit and in order)
2. a regular straight (but not a flush)
3. two of a kind
4. four of a kind
5. two pairs (but not four of a kind)
6. 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:

1. no court cards (J,Q,K,A);
2. at least one ace but no other court cards;
3. at most one kind of court card.

[Pitman p. 128, # 6]