Numb3rs 213: Double Down
The main mathematical topic in this episode are using math to determine
strategies for winning against casinos at blackjack. Along the way, a
discussion of generating psuedorandom numbers comes up.
Counting Cards in Blackjack
Blackjack (also called 21) is a card game which is frequently
played in casinos and is the
central theme of this episode. Since the rules in casinos might be slightly
different than the rules you may have played by in the past, we'll go
through the (slightly simplified) rules first. The goal of the game is
to get cards that add up
to 21 or as close as possible without busting, which means going over
21. Cards with numbers are counted for their number values, face cards (Jack,
Queen, King) are 10 points, and an Ace is either 1 or 11. Each player is only
playing against the dealer (who represents the casino) and not against the
other players. To begin the game, each player puts in a bet and then
the dealer gives each of the players two
cards, face up, and gives herself 1 card face up and another card face down,
as in this example (the dealer is at the top).
Then the player on the dealers left acts. If he is happy with his cards, he
stays, or if he would like another card, he hits. This continues
until he stays or until he busts. Then the rest of the players do the same
thing in turn. Once all the players have had their turn, then the dealer has
her turn. However, the dealer never has the choice of whether to hit or stay,
she must follow predetermined rules. If her hand sums to 16 or less, she must
hit, and if it sums to 17 or more, she must stay.
Activity 1: Odds of the dealer busting
- Let's say you're the dealer and you have a 10 and a 6. You must draw
a card (but you won't have to draw another one). What are the odds that you
bust? (To make it easier assume that the odds of drawing all the different
ranks of cards are the same. That is, you are as likely to draw a 6 as you are
to draw a 7 or 10 or Ace, etc.)
- What if the dealer has cards that sum to 15? What is the probability that
she busts when she draws?
- What if the dealer has cards that sum to 14? What is the probablility
that she busts? (Hint: this is different than the last two, because if she
draws a 2, then her total is 16 and she must draw again.)
Activity 2*: More Odds: Calculate the odds that
the dealer busts for each starting hand of 11 or bigger.
Because of the way the rules are set up, both the players and the casino each
have a different advantage over the other. Since the players don't have to
follow any rules when they hit or stay, they can choose to hit or stay more
optimally, and hence have an advantage. For example, it is usually best not
to hit on 16, but the dealer must every time. However, the dealer has an
advantage because the players must act first, and if the player busts then the
dealer gets the player's money. Very thorough analysis with computers has
resulted in a table that
describes optimal play (this table is only relevent to the full set of rules,
which are also described in the link). If players play according to the table,
then more computer analysis has shown that the casino has approximately a 1
percent advantage, that is, on average, if you bet a dollar, they win one
penny. However, these tables are only based on the current hand, and
when playing at casinos players are able to remember past hands. This knowledge
can give them an advantage.
Since the actual analysis of optimal play is very complex,
either a computer or a whole lot of time is needed to figure it out. Casinos
before the 1960's didn't realize this, and
some of them unintentionally used rule sets there were advantageous for
players who were playing optimally. This changed when the groundbreaking book
"Beat the Dealer: A Winning Strategy for the Game of Twenty-One" by
Edward Thorp was published in 1966.
After completing the activity (or before, with a bit of mathematical intuition)
one sees that if there are many high cards in the deck it is more likely that
the dealer will bust, and if there are few it is more likely that they will
not bust. Since many of the optimal plays in the computer-calculated table
are following the strategy "stay and hope the dealer busts,"
the player has
the advantage when there are many high cards in the deck. A player can use
this to his advantage by remembering one number,
called the count. The count starts at 0, and for every card the player
sees that is worth 10 and every ace the count is increased by one, and for
every card between 2 and 7 the player sees, the count is decreased by 1. Then
if the count is high, there are relatively few high cards in the deck, so the
player is at a disadvantage, and if the count is low, the player has the
advantage. Therefore, when the count is negative players should bet more, and
when the count is positive players should bet less. Using this type of system,
computer analysis has shown that at typical speeds of play, players are
able to win about 1 bet per hour, on average. However, the uncertainty is very
high, so in any particular hour it is not at all unlikely that a player who is
playing optimally (including counting) could win 20 bets or lose 20 bets.
This makes blackjack a poor source of income, unless team play is used.
However, just as the episode indicates, playing with teams can be dangerous.
Here is an extreme example of how
knowledge of previous cards can affect the odds of who wins:
- Repeat activity 2 with the assumption that there are no
cards of value 10 nor any aces in the deck, but that the probability of drawing
any of the other cards is the same.
- Repeat activity 2 with the assumption that there are no aces and no
cards of value 2-7 in the deck, but that the probability of drawing any of
the other cards is the same.
Random Number Generators
Random numbers are useful for many different things. In particular, in this
episode they were used in machines that shuffled cards to make the shuffles
sufficiently random. There are two main ways of producing "random"
numbers. The harder but more effective way is to record some kind of natural
process that is assumed to be random, such as the static on empty radio
stations, and then normalize the data recorded to fix biases in the recording.
An example of this method and more information describing it can be found
A second way to generate "random" numbers is to use a computer
algorithm to produce numbers that satisfy many of the tests that true random
numbers would. Since output from a computer is never random, but is instead
completely determined, numbers generated in this way are called
psuedo-random numbers. One of the simplest ways of doing this is called
a linear congruential generator. This method produces a series of
numbers labelled f(0), f(1), f(2), etc. The first number, f(0), is called
the seed because it has to be generated from an outside source and
it begins the sequence. Then given a number in the sequence f(n), the next
number in the sequence satisfies the equation
Here A,B, and M are fixed numbers selected beforehand. Also, the symbol
(mod M) means "take the remainder after dividing by M".
For example, 27 (mod 10) = 7, 27 (mod 3) = 0, and 27 (mod 4) = 3.
There is a famous example of a very popular linear
congruential generator called
that has been used since the 1960s. The
constants A,B,M were chosen very poorly, and as a result the numbers it
generated were not very random at all. As a result, many simulations done with
this generator are now looked upon with suspiscion.
Testing Random Number Generators:
Now let's try
a couple linear congruential generators and see how well they work. For all
of the following assume that the seed is 1.
- Let's try A = 2, B = 4, M = 10. Write down the first 8 numbers generated.
- Now try A = 3, B = 7, M = 10. Write down the first 8 numbers generated.
- Which of these two choices do you think is better? Why?