none of the 15,000 teams will do this is
(1

1
.
883
×
10

5
)
15
,
000
= 0
.
7539
i.e., with probability 0.2461 some team will have this happen to them.
As a
check on the last calculation, note that (1.13) gives an upper bound of
15
,
000
×
1
.
883
×
10

5
= 0
.
2825
1.5.
BLACKJACK
25
1.5
Blackjack
In this book we will analyze craps and roulette, casino games where the player
has a substantial disadvantage. In the case of blackjack, a little strategy, which
we will explain in this section, can make the game almost even. To begin we
will describe the rules and the betting.
In the game of blackjack a King, Queen, or Jack counts 10, an Ace counts 1
or 11, and the other cards count the numbers that are shown on them (e.g., a
5 counts 5). The object of the game is to get as close to 21 as you can without
going over. You start with 2 cards and draw cards out of the deck until either
you are happy with your total or you go over 21, in which case you “bust.”
If your initial two cards total 21, this is a blackjack, and if the dealer does not
have one, you win 1.5 times your original bet. If you bust then you immediately
lose your bet. This is the main source of the casino advantage since if the dealer
busts later you have already lost.
If you stop with 21 or less and the dealer
busts you win. If you and the dealer both end with 21 or less, then the one with
higher hand wins. In the case of a tie no money changes hands.
In casino blackjack the dealer plays by a simple rule: He draws a card if
his total is
≤
16, otherwise he stops. The first step in analyzing blackjack is to
compute the probability the dealer’s ending total is
k
when he has a total of
j
.
To deal with the complication that an Ace can count as 1 or 11, we introduce
b
(
j, k
) = the probability that the dealer’s ending total is
k
when he has a total
of
j
including one Ace that is being counted as 11.
Such hands are called
“soft” because even if you draw a 10 you will not bust. We define
a
(
j, k
) = the
probability the dealer’s ending total is
k
when he has a hard total of
j
, i.e., a
hand in which any Ace is counted as 1.
We start by observing that
a
(
j, j
) =
b
(
j, j
) = 1 when
j
≥
17 and then start
with 16 and work down.
Let
p
i
= 1
/
13 for 1
≤
i <
9 and
p
10
= 4
/
13.
If
11
≤
j
≤
16 then a new Ace must count as 1 so
a
(
j, k
) =
p
1
a
(
j
+ 1
, k
) +
10
m
=2
p
m
a
(
j
+
m, k
)
When 2
≤
j
≤
10 a new Ace counts as 11 and produces a soft hand:
a
(
j, k
) =
p
1
b
(
j
+ 11
, k
) +
10
m
=2
p
m
a
(
j
+
m, k
)
For soft hands, an Ace counts as 11, so there are no soft hands with totals of
less than 12. If the card we draw takes us over 21 then we have to change the
Ace from counting 11 to counting 1, producing a hard hand, so
b
(
j, k
) =
p
1
b
(
j
+ 1
, k
) +
21

j
m
=2
p
m
b
(
j
+
m, k
) +
10
m
=22

j
p
m
a
(
j
+
m

10
, k
)
When
j
= 11 the second sum runs from 11 to 10 and is considered to be 0.
26
CHAPTER 1.
BASIC CONCEPTS
The last three formulas are too complicated tow rok with by hand but are
easy to manipulate using a computer. The next table gives the probabilities of
the various results for the dealer conditional on the value of his first card. We
have broken things down this way because when blackjack is played in a casino,
we can see one of the dealer’s two cards.
17
18
19
20
21
bust
2
.13981
.13491
.12966
.12403
.11799
.35361
3
.13503
.13048
.12558
.12033
.11470
.37387
4
.13049
.12594
.12139
.11648
.11123
.39447
5
.12225
.12225
.11770
.11315
.10825
.41640
6
.16544
.10627
.10627
.10171
.09716
.42315
7
.36857
.13780
.07863
.07863
.07407