Counting and Gambling
Section 4
Objectives:
• Understand how to use the fundamental
counting principle to analyze counting
problems that occur in gambling.
• Use the theory of permutations and
combinations to compute the probability
of drawing various card hands.
Fundamental Counting Principle
In how many different ways can the three wheels of the slot machine
depicted below stop?
Solution:
Using the fundamental counting principle, we see that the
total number of different ways the three wheels can stop is
20 × 20 × 20 = 8,000.
Fundamental Counting Principle cont. (2)
One payoff for a slot machine with the wheels shown below is for cherries on
wheel 1 and no cherries on the other wheels. In how many different
ways can this happen?
Solution:
By the fundamental counting principle, we see that the total
number of ways we can get cherries only on the first wheel is
2 × 15 × 12 = 360.
Fundamental Counting Principle cont. (3)
The largest payoff for a slot machine with the wheels shown below is for three
bars. In how many ways can three bars be obtained?
Solution:
By the fundamental counting principle, a bar on the first,
second, and third wheel can occur in
2 × 3 × 1 = 6 ways.
Counting and Poker
In the game of poker, five cards are drawn from
a standard 52-card deck. How many different
poker hands are possible?
Solution:
Counting and Poker cont. (2)
• Determine the number of different ways we can obtain four of a kind when
drawing five cards from a standard 52-card deck.
Solution:
Step 1: Pick the rank of the card of which we will have four of a
kind. There are 13 possibilities.
Step 2: Select the fifth card. There are 48 cards remaining.
Total:
13 × 48 = 624 ways.
• How many ways can three cards of the same rank be drawn when drawing
three cards from a standard 52-card deck?
Solution:
Step 1: Pick the rank of the card of which we will have three of a kind.
There are 13 possibilities.
Step 2: Choose three cards from the four cards in the deck with this
rank. This can be done in C(4, 3) = 4 ways.
Total:
13 × 4 = 52 ways
Counting and Poker cont. (3)
Determine the number of different ways we can obtain a full
house when drawing five cards from a standard 52-card deck.
Solution:
Subproblem A: Determine the number of ways we can
choose the three cards having the same rank. There are
52 ways from previous example.
Subproblem B: Determine the number of ways we can
choose the remaining two cards having the same rank.
The rank can be chosen in only 12 ways. Selecting two
of the four cards with this rank can be done in
C(4, 2) = 6 ways.
The two cards can be selected in 12 × 6 = 72 ways.
Total:
52 × 72 = 3,744 ways.
Counting and Poker cont. (4)
If we select five cards from a standard 52-card deck, in how
many ways can we draw a hand with three of a kind?
Solution:
Subproblem A:
Determine the number of ways we can choose
the three cards having the same rank. There are 52 ways from
previous example.
Subproblem B:
Determine the number of ways we can choose
the remaining two cards having the same rank. The fourth card
can be chosen among the 12 remaining ranks in 48 ways, and
then the fifth card can be chosen in 44 ways. To avoid counting
rearrangements of these two cards, we divide by 2. The two
cards can be selected in
ways.
Total:
52 × 1,056 = 54,912 ways