Assignment Instructions & Requirements:
Powerball (Links to an external site.)Links to an external site.is a lottery game which is
a combined large jackpot game and a cash game. Every Wednesday and Saturday
night at 10:59pm Eastern Time, a drawing of five white balls out of a drum with 69 balls
and one red ball out of a drum with 26 red balls is conducted. Players win a prize by
matching one of the 9 Ways to Win (Links to an external site.)Links to an external site..
The jackpot - won by matching all five white balls in any order and the red Powerball - is
either an annuitized prize paid out over 29 years (30 payments counting the first
immediate payment) or a cash lump sum payment.
The Powerball estimated jackpot is currently $109,000,000. That is payable as an
annuity over 30 years or one can receive the net present value of the annuity in a cash
payment ($71,800,000).
1.) What is the jackpot’s expected value to the player who buys a $2 ticket for this
drawing? Please show the expected value for both the jackpot as an annuity and
for the net present value in cash.
For this problem, ignore the other ways to win and focus only on the jackpot. Please
show your work.
Assignment Instructions & Requirements:
Notes:
•
•
•
Any mention of a deck of cards refers to a standard 52 card deck, composed of four
suits (♠ ♣ ♥ ♦) where (♠ ♣) are black and (♥ ♦) are red. Within each suit cards are
ordered 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A, with J, Q, K, A considered face cards.
Problems involving dice refer to standard 6 sided die and that we observe the face
of the die facing upwards.
All coins tossed are fair coins with two sides (heads and tails).
Answer on a separate, typed document. Please show all necessary work in answering
the problems.
1. From a shuffled deck of cards, what is the probability of drawing a face card that is a
♣? What is the probability of drawing a face card or a ♣?
2. When rolling a pair of dice, what is the probability that they will show a total of 8?
3. What is the probability of flipping a coin and observing three consecutive heads?
4. From a single deck of cards, assume that the following cards have already been
played: J♦, 4♦, 10♥, A♥, 8♣, 5♣, 3♣, 7♠, 9♠. What is the probably of drawing a 4 or
less on the next card? What is the probability of drawing a face card on the next
card?
5. When rolling a pair of dice, what is the probability of rolling at least one seven in four
rolls?
Gambling Math
GAMING & CASINO MANAGEMENT
Gambling Math
• How are Card Games like Blackjack different from
games such as Roulette, Craps & Slots?
• Roulette, craps and slots have elements of independence and
randomness throughout
• Card games odds are constantly changing due cards drawn
• IE – cards drawn in the first hand will affect odds on subsequent hands
Independent vs Dependent
• Independent
• Coin Flips
• Dice
• Roulette
• Future events not based
on past/current
outcomes
• Dependent
• Cards
• Previous hands of cards
•
impact future hands
Once a card is dealt, it can
never be dealt again
Gambling Math
Blackjack
• Keeping track of which cards
have been played can give the
player clues as to which cards
may come next
• This does not guarantee
success, but allows the
player to utilize strategy
Roulette or Craps
• The wheel and the dice have
no memory; keeping track of
previous rounds provides no
advantage
Roulette Winners
Casino Games
• When the player enters the casino, she is faced with options:
•
Inter-Game Options
•
•
What Game to play (ie – slots, blackjack, poker, roulette)
Intra-Game Options
• What game or table to play
• In game options for betting (ie – win, place, show bets for horses)
• Understanding the odds gives the player a better chance and
possibly positive expected values
Probability
Gambling Math
Quick Facts on Probability
• If something cannot happen (impossible),
it has a P = 0
• If something must happen (certainty),
it has a P = 1
• Probability of occurrence X is referred to as the P(X)
HEADS
TAILS
Quick Facts on Probability: Coin Toss
• A Coin Flip:
• 50% Chance of Heads
• 50% Chance of Tails
• Probability of Heads =
• P(Heads) =
1
2
= .5
Heads Outcomes
Total Outcomes
Quick Facts on Probability: Dice
• A Single Die has 6 sides, numbered 1,2,3,4,5,6
• Probability of any number is 1/6 or .167
P(1) =
P(2) =
P(3) =
P(4) =
P(5) =
P(6) =
.167
.167
.167
.167
.167
.167
1.000
Games are played with Two Dice, not a single die,
so the probability is based on:
6 X 6 = 36 Outcomes
Dice Probability Examples
• What is the Probability of getting a 7 when
rolling two Dice?
• For this, we need to know:
P(7) =
Number of Ways to Get 7
Total Outcomes
(1,6) (2,5) (3,4) (4,3) (5,2) (6,1)
P(7) =
36
6
=
36
1
=
6
=
.167
Two Dice Outcomes & Probabilities
Outcome
X/36
Probability
Roll a 2: (1,1)
1/36
0.0277778
Roll a 3: (1,2) , (2,1)
2/36
0.0555555
Roll a 4: (1,3), (2,2), (3,1)
3/36
0.0833333
Roll a 5: (1,4), (2,3), (3,2), (4,1)
4/36
0.1111111
Roll a 6: (1,5), (2,4), (3,3), (4,2), (5,1)
5/36
0.1388889
Roll a 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
6/36
0.1666667
Roll a 8: (2,6), (3,5), (4,4), (5,3), (6,2)
5/36
0.1388889
Roll a 9: (3,6), (4,5), (5,4), (6,3)
4/36
0.1111111
Roll a 10: (4,6), (5,5), (6,4)
3/36
0.0833333
Roll a 11: (5,6), (6,5)
2/36
0.0555555
Roll a 12: (6,6)
1/36
0.0277778
Quick Facts on Probability: Cards
• A Standard Deck of Cards:
• 52 Cards, assumed random, shuffled
• Four Suits: Hearts, Diamonds, Clubs, Spades
• Aces, 2-10, Jack, Queen, King
STANDARD DECK OF 52 CARDS
Quick Facts on Probability: Cards
• Typical Probabilities
• Probability of any one card 1/52 = .019
• Probability of getting an Ace = 4/52 = 1/13 = .077
• Probability of getting a Spade = 13/52 = ¼ = .25
Expected Value
Gambling Math
Expected Value introduces Betting Odds
to Probability
If we know the probability of an outcome and
the amount we wager and win in a bet, we can
determine the Expected Value
Expected Value
E(X) = [P(W) x Amount Won] + [P(L) x Amount Wagered]
Probability
of Winning
Dollar Amount
of Gain in Win
Probability
of Losing
Dollar Amount
of Loss
(THIS IS A NEGATIVE
DOLLAR AMOUNT)
Mathematical Expectation
• The Mathematical Expectation for any game
is as follows:
• Multiply each possible gain or loss by the
probability of that gain or loss
• A “Fair Game” is one with an Expected Value
of 0
• Neither the house nor player have an advantage
Mathematical Expectation
•An Example of Coin Toss
• We know an unbiased coin, over time, will deliver
50% heads and 50% tails
• If given a chance to bet even money ($1 bet = $1 win)
on next toss
• Mathematical Expectation in this example is:
(.5)(1) + (.5)(-1) = 0
A Positive Expected Value Example
• Suppose for our next coin toss, we’re offered 3/2 odds,
that is a gain of $1.50 for our $1.00 bet
• What is our formula?
(.5)(1.50) + (.5)(-1) = +.25
• This game has a positive expected value of .25
• If we played this game 100 times, we’d expect to win $25
Blackjack Expected Value Example
ODDS IN BLACKJACK
• Object of Blackjack: Closest to 21 without going over
• First hand of a single deck blackjack game. Deck is assumed
to be shuffled and random.
• Player Bets $100 and is dealt a 10, K (equaling 20)
• The Dealer shows a 10
• In the deck there are now 49 cards including
• 13 Tens (K, Q, J, 10) and
• 32non-tens (2, 3, 4, 5, 6, 7, 8, 9)
• 4 Aces
• What are the odds that the Dealer wins?
ODDS IN BLACKJACK
• What are the odds that the Dealer wins?
• First, if we have 10+10 and the Dealer has 10 showing, what
card does the Dealer need to beat us?
ODDS IN BLACKJACK
• What are the odds that the Dealer wins?
• The Dealer Needs an Ace
• Ace has a Value of 11
• 11 + 10 = 21 > 10 + 10 = 20
• So, we want to find the probability of the Dealer holding
an ACE
ODDS IN BLACKJACK
• So, we want to find the probability of the Dealer holding
an ACE
• We know there are three Tens showing, so in the decks
there are now 49 cards including
• 13 Tens (K, Q, J, 10) and
• 32 non-ten numbered cards (2, 3, 4, 5, 6, 7, 8, 9)
• 4 Aces
CARD VALUES IN BLACKJACK
1
or
11
Face Value of the Card
i.e. 2,3,4,5,6,7,8,9
10
A Note on ACE cards in Blackjack:
An ACE can either have a value of 1 or 11,
based on the needs of the player.
In the following examples, we will treat the ACE as 11.
ODDS IN BLACKJACK
• So, we want to find the probability of the Dealer holding
an Ace:
P(ACE) =
Number of Ways to Get Ace
Total Outcomes
=
4
49
=
.0816
=
.9184
• And the dealer not holding an Ace:
P(NOT) =
Number of Ways Not Ace
Total Outcomes
=
45
49
ODDS IN BLACKJACK
• So, we want to find the probability of the Dealer holding
an Ace:
P(ACE) =
Number of Ways to Get Ace
Total Outcomes
=
4
49
=
.0816
Notice
Adding
these = 1
• And the dealer not holding an Ace:
P(NOT) =
Number of Ways Not Ace
Total Outcomes
=
45
49
=
.9184
ODDS IN BLACKJACK
• Now that we know the P(ACE) and P(NOT), we can
determine the Expected Value of this Bet:
E(X) = (.9184)($100) + (.0816)(-$100) =
E(X) = ($91.84) + (-$8.16) =
E(X)) = $83.68
The Bad News:
This Bet doesn’t exist in Blackjack.
You place your wager before you know what cards you’ll receive
Let’s do a Real-World Example of
Expected Value in Blackjack.
The Insurance Bet
EXPECTED VALUE IN BLACKJACK
• First hand of a 4 deck blackjack game. Deck is shuffled and random
• Player Bets $12 and is dealt a 6, 5
• In the 4 decks there are now 205 cards including
•
•
64 Tens (K, Q, J, 10) and
144 non-tens (2,3,4,5,6,7,8,9,A)
• Dealers Shows ACE and offers insurance
• Insurance is a $6 bet that pays $12 if Dealer has a 10 or $6 otherwise
• The players EV on the Insurance Bet is:
(64/205)(12) + (141/205)(-6) = -78/205 or -.3805
EXPECTED VALUE IN BLACKJACK
CONTINUED
(Tens/Remaining Cards)(Amount of Bet) + (Non-Tens/Remaining Cards)(Amount of Loss)
(64/205)(12) + (141/205)(-6) = -78/205
(.3122)(12) + (.6878)(-6) =
3.7464 + -4.1268 = -0.3804
So what does this mean?
-$.38 is a negative EV for the Player and
thus the Player should decline insurance
NOTE: Remaining Cards = 208 Total – (Player’s 2 cards, Dealer’s 1 Up Card)
Expected Value of Dice
A Craps Example
EXPECTED VALUE OF A DICE GAME
Similar to a Pass Line Bet in
Craps, Joe is betting $10 that
the roll of two dice does not
equal a 2, 3 or 12
(32/36)($10) + (4/36)(-$10)
(8.889) + (-1.111) = $7.778
Outcome
X/36
Probability
Roll a 2: (1,1)
1/36
0.0277778
Roll a 3: (1,2) , (2,1)
2/36
0.0555555
Roll a 4: (1,3), (2,2), (3,1)
3/36
0.0833333
Roll a 5: (1,4), (2,3), (3,2), (4,1)
4/36
0.1111111
Roll a 6: (1,5), (2,4), (3,3), (4,2), (5,1)
5/36
0.1388889
Roll a 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
6/36
0.1666667
Roll a 8: (2,6), (3,5), (4,4), (5,3), (6,2)
5/36
0.1388889
Roll a 9: (3,6), (4,5), (5,4), (6,3)
4/36
0.1111111
Roll a 10: (4,6), (5,5), (6,4)
3/36
0.0833333
Roll a 11: (5,6), (6,5)
2/36
0.0555555
Roll a 12: (6,6)
1/36
0.0277778
EXPECTED VALUE OF A DICE GAME
What if Joe were to bet
“Don’t Pass” instead where he
wins on a 2, 3 or 12?
(4/36)($10) + (32/36)(-$10)
(1.111) + (-8.889) = -$7.778
Outcome
X/36
Probability
Roll a 2: (1,1)
1/36
0.0277778
Roll a 3: (1,2) , (2,1)
2/36
0.0555555
Roll a 4: (1,3), (2,2), (3,1)
3/36
0.0833333
Roll a 5: (1,4), (2,3), (3,2), (4,1)
4/36
0.1111111
Roll a 6: (1,5), (2,4), (3,3), (4,2), (5,1)
5/36
0.1388889
Roll a 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
6/36
0.1666667
Roll a 8: (2,6), (3,5), (4,4), (5,3), (6,2)
5/36
0.1388889
Roll a 9: (3,6), (4,5), (5,4), (6,3)
4/36
0.1111111
Roll a 10: (4,6), (5,5), (6,4)
3/36
0.0833333
Roll a 11: (5,6), (6,5)
2/36
0.0555555
Roll a 12: (6,6)
1/36
0.0277778
Expected Value of the Lottery
EXPECTED VALUE OF A LOTTERY GAME
Jim loves to play the lottery, but only when the
jackpot is over $350,000,000, which is the mark it just
hit today. Jim would elect take the cash payment, as
opposed to the annuity payments, which would
reduce the jackpot payout to $260,000,000.
Jim is buying his $1 ticket, but is curious about the EV
of this game (jackpot only).
(1/258,890,850)($260,000,000) + (258,890,849/258,890,850)(-1) =
(.0000000038626)($260,000,000) + (.9999988373479)(-1) =
(1.0032842379327) + (-.9999988373479) = .0042854005848
The expected value is $0.00
These are the chances of winning the New York Mega Millions from nylottery.ny.gov
IMPORTANCE OF REPETITION
• It’s important to note that EV is the amount you will tend
to win or lose on average over the course of repeated trials
• Randomness plays a critical role in individual trials
• Even Money Wager; $100 bet per toss; 10 Tosses
• Randomness at play in 10 Tosses; over 1,000 or more
tosses we’d see the trend to be even because it’s a Fair
Game
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