Name: ___________________________________
Date: ____________________
Intro to Stats
Winning at Blackjack
A standard deck of playing cards has 52 cards. There are four suits (clubs,
diamonds, hearts, and spades), each of which has thirteen numbered cards (2,
..., 9, 10, Jack, Queen, King, Ace).
In the game of blackjack, each card is worth an amount of points. Each
numbered card is worth its number (e.g., a 5 is worth 5 points); the Jack,
Queen, and King are each worth 10 points; and the Ace is either worth your
choice of either 1 point or 11 points. The object of the game is to have more
points in your set of cards than your opponent without going over 21. Any set
of cards that sum greater than 21 automatically loses.
Here's how the game is played. You and your opponent are each dealt two
cards. Usually the first card for each player is dealt face down, and the second
card for each player is dealt face up. After the initial cards are dealt, the first
player has the option of asking for another card or not taking any cards. The
first player can keep asking for more cards until either he or she goes over 21, in
which case the player loses, or stops at some number less than or equal to
21. When the first player stops at some number less than or equal to 21, the
second player then can take more cards until matching or exceeding the first
player's number without going over 21, in which case the second player wins,
or until going over 21, in which case the first player wins.
We're going to simplify the game a little and assume that all cards are dealt
face up, so that all cards are visible. This is a wimpier game than the face-down
one, but it makes for easier probability calculations!
Name: ___________________________________
Date: ____________________
Intro to Stats
Problems:
In all these questions, assume your opponent is dealt cards and plays first.
1. What is the chance that the first card will be a heart and a Jack?
2. What is the chance that the first card will be a heart or a Jack?
3. Given that the first card is a heart, what is the chance that it will be a Jack?
4. Given that the first card is a Jack, what is the chance that it will be a heart?
5. Your opponent is dealt a King and a 10, and you are dealt a Queen and a =
8. Being smart, your opponent does not take any more cards and stays at
20. What is the chance that you will win if you are allowed to take as many
cards as you need?
Name: ___________________________________
Date: ____________________
Intro to Stats
Answers
There are 52 cards in a deck, so that the total number of outcomes for the first
card is 52. Each is equally likely to be picked.
(i) There is only one way to get a Jack and a heart: get the Jack of
hearts. Hence, P(Jack and heart) = 1/52.
(ii) There are 16 ways to get a Jack or a hearts: get one of the thirteen hearts
(Ace through King of hearts), or get one of the Jack of clubs, Jack of spades, or
Jack of diamonds. Hence, P(Jack or hearts) = 16/52. Notice that the Jack of
hearts is counted in the thirteen hearts cards, so that I didn't count it again
when listing the Jacks.
(iii) There are thirteen hearts. There is only one Jack among those 13 hearts:
the Jack of hearts. If we know the first card is a hearts, then the chance that it
is a Jack is the number of ways to get the Jack of hearts out of the total
number of hearts. Hence, P(Jack | hearts) = 1/13.
(iv) There are four Jacks. There is only one heart among these four Jacks: the
Jack of hearts. If we know the first card is a Jack, then the chance that it is a
heart is the number of ways to get the Jack of hearts out of the total number
of Jacks. Hence, P(Jack | hearts) = 1/4.
Name: ___________________________________
Date: ____________________
Intro to Stats
(v) To win, you have to get a 21. There are several ways to get 21: draw a three
in one card, draw a two and an Ace in two cards, and draw three Aces in three
cards.
Getting 21 by drawing a three:
Since there are four threes in the deck, and 48 cards remaining after the first
four cards are dealt, the chance of getting a 3 is P(get a 3) = 4/48.
Getting 21 by drawing an Ace first and a 2 second:
P(get Ace first and a 2 second) = P(get a 2 second | get an Ace first) * P (get an
Ace first)
= (4 / 47) * (4/48)
The 4/48 is determined as follows. Since there are four aces in the deck, and 48
cards remaining after the first four cards are dealt, the chance of getting an
Ace first is 4/48.
The 4/47 is determined as follows. Since there are four 2s in the deck, and 47
cards remaining after the first five cards are dealt, the chance of getting a 2 on
the next card is 4/47.
Getting 21 by drawing a 2 first and an Ace second.
By similar logic,
P(get 2 first and Ace second) = P(get Ace second | get 2 first) * P(get 2 first)
= (4 / 47) * (4/48)
Getting 21 by drawing three Aces.
There are 4*3*2 = 24 ways to get three aces, and there are (48*47*46) ways to
pick three cards. Hence, the probability of picking three aces is 24/(48*47*46).
Hence, P(win) = 4/48 + (4*4)/(47*48) +
(4*4)/(47*48) + (4*3*2)/(47*48*46) = .0907.