Transcript Lottery Lecture 2014 (April 11 version)

welcome to…
“A Lotto Learning: Lottery Literacy”
with Dr. Lawrence (Larry) Lesser
Professor, Department of Mathematical Sciences
Interim Director, Center for Excellence in Teaching and Learning(CETaL)
With a little math (and psychology), make a more informed choice
on how (or whether) to play lotteries like LottoTexas!
Texas connections
• Education: Houston public schools,
Rice (BA in Math) &
UT-Austin (MS in Stats; PhD in Math Ed)
• Work in TX: includes lecturer (UT-Austin,
Southwestern U., St. Edward’s U.),
HS teacher, state agency statistician,
professor at UT-El Paso (since 2004)
2+ decades of lottery trajectory
• Nov. 14, 1992: Texas Lotto begins
while I’m a math ed PhD student at UT-Austin
• 1993: my adult ed class at UT (covered by
media all the way up to CNN Headline News!)
• 1997: my 1st math ed journal article was on
using the lottery to teach statistics
• 2006: I reprise my adult ed class at UTEP
• 2014: UTEP Centennial Open House Lecture
Lottery Coverage in the Media
• Newspapers: El Paso Times, Houston Chronicle,
Austin Chronicle, Austin Business Journal,
Austin American-Statesman, Daily Texan, etc.
• Nat’l magazines: Real Simple, BottomLine Retirement
• Radio: El Paso, San Antonio, Austin, Houston, Atlanta
• TV: El Paso(KVIA, KFOX), Austin (KVUE), CNN
• Internet: website, YouTube video*
*1st-place award in 2011-12 ‘Quantitative Literacy in the Media’ contest sponsored by QL-SIGMAA, and
“Best Online Submission” prize in 2014 ‘ASA’s Got Talent’ contest sponsored by American Statistical Association
my adult ed class
(Austin newsstory ran 37 column inches and
led to coverage by CNN Headline News!)
Media: Lessons learned
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Think how to connect discipline to society
Know audience/format
Start and end with main point
Have “sound bites” ready
Concrete analogies from everyday experience
Ask to accuracy check write-up
Redirect/reframe question when necessary
Speaking of the media,
what headline(s) have you seen?
a) “Stats Prof Wins Lotto Again!”
b) “Psychic Wins Lotto Again!”
c) both of the above
d) none of the above
good reasoning or
“equiprobability bias”?
Quiz
What is the probability of spinning RED?
a) 1/2 b) 1/3 c) 1/4 d) none of the above
What are the odds in favor of RED?
a) 1:2 b) 1:3 c) 1:4 d) none of the above
Quiz
What is the probability of spinning RED?
a) 1/2 b) 1/3 c) 1/4 d) none of the above
What are the odds in favor of RED?
a) 1:2 b) 1:3 c) 1:4 d) none of the above
and what are the odds against RED? 3:1
NOTE: Texas Lottery (mis)uses the word “odds” as if
it means “probability”.
Today’s context…
is LOTTO TEXAS,
which draws 6 balls (from 1-54)
Wed. & Sat. nights
where Texas Lottery $ goes
http://www.txlottery.org/export/sites/lottery/Supporting_Education/index.html
Financial literacy followups
(before students can legally buy ticket: 18 in most states, including TX)
• choose (as TX makes you up front):
annuity (30 annual payments each 1/30 of the jackpot)
or
lump sum (cash value option) ≈ 1/2 jackpot
(consider assumed interest rate, tax rate, financial needs, etc.)
• Does TX total education spending increase
by full amount of earmarked lottery revenue?
some economics/psychology
• You’re ____ likely to play after a “near-miss.” A) more B) less
Gamblers transform their losses into “near wins”. “Sunk cost bias”.
• Lottery ticket buying goes ____ after a jackpot rollover.
A) up B) down
“halo effect”
• Lottery ticket buying is ____ correlated with size of the
jackpot prize. A) positively B) negatively C) not
• Lottery ticket buying is ____ correlated with level of
education. A) positively B) negatively C) not
“Lottery: a tax on people who are bad at math” – American writer Ambrose Bierce (1842-1914)
• Low-income people spend ___ % of their income on the
lottery than others. A) higher B) lower C) same
“lottery is regressive”
probability of Lotto Texas jackpot
1
25,827,165
≈ 1 in 26 million
How to Choose Things:
verifying the 1 in 26 million
• How many ways(“combinations”) can you
pick a 2-person committee from 4 people (A,
B, C, D)? Hold up 4 fingers
• C(n,k) ways to choose k items from n items;
𝑛!
“n choose k” =
,
𝑘! 𝑛−𝑘 !
where n! is product of first n positive integers (so 4! = 4x3x2x1 = 24)
• Lotto Texas (n = 54, k = 6):
54!
6!48!
(can also verify via the hypergeometric distribution)
= 25,827,165
Jackpot probability tradeoffs
too small: jackpot wins rare
too large: rollovers rare,
so big jackpots rare
“just right”: about 1 in
(state population size)
but how to visualize 1 in 26 million?
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a minute in a half-century
an inch on the road from El Paso to Phoenix
a 5-foot segment of the equator
a sheet of typing paper from a 1.6-mile stack
a square inch from an area bigger than
3 football fields (with endzones)
• a person from 500 full Sun Bowls!
• buy 100 tickets/week and on average you win
once every 5,000 years
for perspective, some events
as (or more) likely than that jackpot:
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Becoming President of the USA
Becoming a movie star
Having identical quadruplets
Being struck by lightning
Death by airline-related terrorist attack
Death by bee/hornet/wasp sting
Death by flesh-eating bacteria
Dying in a bathtub
Death by car accident while driving 1 mile
“Your chances of winning the lottery are pretty much the same whether
you buy a ticket or not” – journalist Dan Rather
2 ways to state likelihood
Relative: buy a second ticket and
your chance of winning is doubled!
Absolute: buy a second ticket and
your probability of winning goes from
.00000003872 to
.00000007744 !
Quick….
call out a reason
people play
some reasons people play
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entertainment/daydream like a matinee
respite from financial anxiety
such a small investment – $1 is “nothing”
quit job, support family, pay medical bills
credibility: it’s backed by the government
“someone must win – why not me?”
everyone has same odds (unlike life)
only imaginable path to multi-millionaire
(vs. real estate or stock market)
• philanthropy (e.g., help fund TX education)
Classroom Story
Teacher says, “OK class, write an essay on
what you’d do with $10 million from the lottery.”
Joe hands in a blank sheet and is asked by the
teacher why he didn’t write anything.
Joe answers, “If I had $10 million,
that’s exactly what I would do: nothing!”
Reflect:
What would YOU do
if you won? Really.
Quick….
call out a reason
people don’t play
some reasons people don’t play
• no state lottery (AK, AL, HI, MS, NV, UT, WY)
• no $ beyond food, rent, and clothing
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privacy/security fears
odds too small (essentially 0)
a religious view of gambling
view that $ won’t ensure  (or even $!)
exploits gambling addicts, impulse buyers
exploits poor; increases wealth inequality
player loses $ on average
Expected Value(EV) calculation
You have a 1/5 chance of winning $100
and a 4/5 chance of winning $10.
What is EV of your winnings?
• On average, if you play 5 times, you win
$100, $10, $10, $10, $10.
$140 for 5 plays is $28 per play
• Or add the “probability × payoff” pieces:
1
4
($100) + ($10) = $28
5
5
QUIZ
When you pay $1 for a lottery ticket,
about how much of that $1 are you losing,
on average, to the nearest 10¢?
A) 10¢
B) 30¢
C) 50¢
D) 70¢
E) 90¢
a $1 Lotto Texas ticket….
# of balls matched
probability
typical prize
6 (all of them)
1/25,827,165 $10 million
5
1/89,678
$2,000
4
1/1,526
$50
3
1/75
$3 (always)
Comparing games of chance
Lotto ticket returns 48% of ticket price to player
For comparison….
• American roulette returns 36/38 = 95%
• Blackjack returns about 99%
• Craps returns about 99%
Purchases with negative net EV
• Lottery tickets
• Health insurance
• Extended warranty on an appliance
STRATEGIES in Baldo comic?
http://www.txlottery.org/export/sites/lottery/Games/Lotto_Texas/Winning_Numbers/
Tracking (some call it “locking”)
Suppose “7” is the only number that was drawn
in each of the last 3 drawings. It is _____ any
other number to occur in the next drawing.
A) more likely than
B) less likely than
C) equally as likely as
C is correct,
but people pick as if B is correct
Suppose “7” is the only number that was drawn
in each of the last 3 drawings. It is _____ any
other number to occur in the next drawing.
A) more likely than (“hot hand”)
B) less likely than (“law of averages”; gambler’s fallacy)
C) equally as likely as (the balls have no memory!)
Austin Chronicle ad
5/7/1993, p. 61
my response in the next issue
Independence
• Each draw is not affected by others
• Assessed with pre-tests on sets of balls
• Events A & B are independent means:
P(A and B) = P(A)×P(B)
P(A given B) = P(A)
P(B given A) = P(B)
from www.beatlottery.net/how-to-win-the-lottery
• “For some reason 1 number from previous
draw is often being drawn in the next
draw. So when picking your numbers you
may pay attention to one of the numbers
drawn in the last draw.”
• My calculation for Lotto Texas:
48
47
46
45
44
43
Pr(no repeats) = 54 × 53 × 52 × 51 × 50 × 49 = 47.5%
So there is at least one repeat 52.5% of the time!
People tend to underestimate probabilities of “something
happening at least once” (Bar-Hillel & Neter, 1993; Shaughnessy 1977)
and some people track SETS of numbers….
(from http://www.beatlottery.net/how-to-win-the-lottery)
“Don’t play combinations that have been drawn
before! Till this date none* of the winning
combinations repeated! Playing them will most
likely guarantee you that you will never win a
jackpot again.”
* not sure if true for all lotto games worldwide,
but it’s still irrelevant
What’s more likely to win jackpot?
A) {1, 2, 3, 4, 5, 6}
B) {11, 19, 28, 37, 40, 51}
C) A & B are equally likely
What’s more likely?
A)
The winning set of numbers
consists of 6 consecutive numbers
B) The winning set of numbers
includes no consecutive numbers
C) A & B are equally likely
specific combo ≠ category of combos
Real Simple interview:
and a professor in UTEP’s
Dept. of Mathematical Sciences
June 2005 Real Simple, p. 41
using Gail Howard’s approach…
Suppose you’re picking numbers from 1-50:
She might say to avoid 1-10 because those
numbers come up 10/50 = 20% of the time,
while 11-50 come up 80% of the time.
continuing that logic…
1-10 come up only 20% of the time,
11-20 come up only 20% of the time,
21-30 come up only 20% of the time,
31-40 come up only 20% of the time,
41-50 come up only 20% of the time.
So avoid everything! 
You don’t pick a category, you pick 6 numbers
and all 6-number combos are equally likely!
Conclusions on Tracking
• What effect will tracking have on your
probability of winning?
• What effect will tracking have on the
expected value of your winnings?
Conclusions on Tracking
• What effect will tracking have on your
probability of winning? NONE!
• What effect will tracking have on EV of
your winnings? NONE!
(if anything, it might reduce it because if you
do win, you’ll have to share with all those
using that strategy!)
what is WHEELING?
• Pick set of at least 7 numbers: e.g., {1,2,3,4,5,6,7}
• Buy 1 ticket for each 6-number subset:
TICKET #1
TICKET #2
TICKET #3
TICKET #4
TICKET #5
TICKET #6
TICKET #7
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
• Note: # of combos increases fast! To buy
all 6-number combos from 1-10 takes $210!
Conclusions on Wheeling
• What effect will wheeling have on your
probability of winning the jackpot?
• What effect will wheeling have on EV of
your ticket?
How does wheeling affect….
• …probability of winning the jackpot?
same as ANY set of 7 nonidentical tickets
• …probability of winning any amount? Fewer
numbers are used, so chance of any kind of
win is lower, but IF you win, you’ll win more.
(Some wheeling systems can guarantee
winning a prize, but the math is hard, and a
huge number of tickets is needed.)
• …EV of your ticket? No change.
Pooling – what is it?
• Form a group of, say, 10 trusted friends.
• Each of you buys a ticket.
• Each of you has an equal share of all 10
tickets, so you get 1/10 of the winnings
from any winning ticket.
Gail Howard’s tips on pooling
www.smartluck.com/free-lottery-tips/texas-lotto-654.htm
“A jackpot could happen because of the luck
just one member brings to your pool.
Select your partners carefully…Avoid
negative people. Not only are they unlucky,
but they dampen enthusiasm and drain
energy from others…
One quick way to tell winners from losers is
simply to ask them: ‘Do you think you are a
lucky person?’”
Conclusions on Pooling
• What effect will pooling have on your
probability of winning the jackpot?
Increased by factor of 10 (as you would by
buying ANY set of 10 nonidentical tickets)
• What effect will pooling have on EV of
your ticket?
If you increase your chances by factor of 10
but only get 1/10 of any winnings, your EV of
winnings per $ spent remain the same.
So why pool?
If the jackpot is $10 million,
how would you prefer to spend $1?
A) Go “solo” and have
1 chance of winning $10 million
B) Form a 10-person pool and have
10 chances of winning $1 million
So, having discussed tracking, wheeling,
and pooling, is there any strategy...
• to increase chance of winning per $1 spent?
NO, unless you join a pool
• to increase EV of winnings IF you win?
YES, if you avoid combinations played by
more people
what numbers do people often pick?
& numerical/visual patterns, “due” numbers, etc.
forms of lottery outreach I’ve done
• adult ed. courses (UT-Austin, UT-El Paso)
• pieces in 5 education journals:
J. of Statistics Education
March 2013
Mathematics Teacher
Sept. 2012
Statistics Teacher Network
Winter 2004
Texas Mathematics Teacher
Fall 2003
Spreadsheet User
Nov. 1997
• TV/radio/magazine interviews
• award-winning YouTube video
• Lottery Literacy webpage
http://www.math.utep.edu/Faculty/lesser/lottery.html
• and a SONG!
my lottery education song!
What #1 country hit
has a title begging
to be made into a
lottery education song?
(and what are the odds
that I have a guitar to play it now?
I need volunteer to advance slide)
“The Gambler” © 2001, 2009 L. Lesser
On a warm summer’s evenin’,
on a train bound for nowhere,
I met up with a gambler -we were both too tired to sleep.
So he told me how he planned
winnin’ lottery prizes
‘Til, as a math teacher, I just had to speak:
“Son, you track those draws,
you say ya got a system–
You call some numbers “hot”,
you deem others “due”;
But I insist,
they each have the same chance–
If you’re gonna play the game, boy,
Ya gotta know what’s true!”
“The Gambler” © 2001, 2009 L. Lesser
SING ALONG: You gotta
Know when you pick ‘em,
What’s superstition,
Know what is strategy
And know when there’s none!
You never try to learn this
At the 7-11:
Take the time right now for learnin’
When the singin’s done!
[now, watch my capo do a math-&-music “translation”…]
“The Gambler” © 2001, 2009 L. Lesser
Now all sets of numbers
are equally unlikely,
More rare than death by lightning,
still there’s somethin’ you should know;
If you should happen
to win that big jackpot,
You’ll win more money
if you picked it all alone!
So avoid those numbers
that more folks are playin’:
Like 7’s and birthdays
and sequences, too.
‘Til this song gets famous,
you’ll have the advantage–
Maybe you’ll thank me
with a share of your loot!”
“The Gambler” © 2001, 2009 L. Lesser
SING ALONG : You gotta
Know when you pick ‘em,
What’s superstition,
Know what is strategy
And know when there’s none!
You never try to learn this
At the 7-11:
Take the time right now for learnin’
When the singin’s done!
[now, watch my brief ‘steel guitar solo’….]
Thanks for coming today!
May the odds be ever in your favor!
Professor Lesser
http://www.math.utep.edu/Faculty/lesser/lottery.html
QUESTIONS?
Other things I can discuss if time/interest:
1.)“Trump Ticket” strategy (buying ALL combos)
2.)1 10-ticket drawing vs. 10 1-ticket drawings?
3.)How likely is some adjacent numbers?
4.)How likely is a “near miss”?
5.) “all even or all odd” example
6.)$1 Lotto Texas vs. $2 Lotto Texas Extra?
1.)What if you bought
all (26 million different) combos?
GOOD NEWS:
You win (a share of) the jackpot! 
BAD NEWS:
• You are not guaranteed a profit. 
• Logistics of buying all those tickets in time. 
2.)What’s better?
A) Buy 1 ticket per drawing for, say, 10 drawings
B) Buy 10 (different) ticket combos for 1 drawing
C) No difference
Which is better?
A: Buy 1 ticket per drawing for 10 drawings:
Pr(at least 1 jackpot) = 1 – (1 – (1/N))10
B: Buy 10 different ticket combos for 1 drawing:
Pr(jackpot) = 10/N
Binomial inequality (1+x)n ≥ 1 + nx
with x = -1/N and n =10 shows
higher probability for option B;
but EV of winnings is the same
A simplified example: choosing
one number from {1,2,3,4,5}
• Method 1: buy 2 tickets in 1 drawing
You have 2/5 = 40% chance of a win
• Method 2: buy 1 ticket in each of 2 drawings
You have 2 wins (1/5)(1/5) = 4% of the time
You have 1 win (1/5)(4/5)+(4/5)(1/5) = 32% of the time
So probability of at least 1 win is 36%,
but there’s the same expected # of wins: .04*2 + .32*1 = .40
3.)how likely is getting some adjacent numbers
in a draw such as {5, 14, 15, 26, 31, 40}
Pr(some adjacent numbers)
= 1 – Pr(no adjacent numbers)
= 1–
𝐶(𝑛−𝑘+1, 𝑘)
=
𝐶(𝑛,𝑘)
1–
𝐶(49,6)
≈
𝐶(54,6)
.46
so it happens almost half the time!
4.)A “near-miss” example
If {5, 14, 23, 32, 41, 50} are the winning
numbers, what’s the probability that each of
your chosen numbers is within ___ of some
member in the above set?
1  (C(18,6)-1)/C(54,6) = 0.1%
2  (C(30,6)-1)/C(54,6) = 2%
3  (C(42,6)-1)/C(54,6) = 20%
5.)from www.beatlottery.net/how-to-win-the-lottery
• “Don’t play all ODD or all EVEN numbers!
Chances of winning the jackpot this way
are extremely low. Best mix is even
selection of odd and even numbers”
• My calculation for Lotto Texas:
Pr(x odd numbers) = C(27, x) * C(27, 6-x) / C(54, 6)
IS greatest for x = 3, but this is useless in telling you what specific 6ball set to pick; every 6-ball combo is still equally likely, regardless
of how many of its numbers are odd
from www.beatlottery.net/how-to-win-the-lottery
“Don’t play all ODD or all EVEN numbers!”
CONSIDER: simplified example of choosing
2 numbers out of {1,2,3,4}
Both even:
Both odd:
Half even, half odd:
from www.beatlottery.net/how-to-win-the-lottery
“Don’t play all ODD or all EVEN numbers!”
CONSIDER: simplified example of choosing
2 numbers out of {1,2,3,4}
Both even: {2,4}
Both odd: {1,3}
Half even, half odd: {1,2}, {1,4}, {2,3}, {3,4}
But you pick a pair, not a category, and all 6 pairs are equally likely.
6.)
a $2 Lotto Texas Extra ticket….
# of balls
matched
probability
Added value
over $1 ticket
all 6
1/25,827,165
none
5
1/89,678 = .000011
$10,000
4
1/1,526 = .000655
$100
3
1/75 = .013
$10
2
2,918,700 / 25,827,165 = .113
$2
has EV of 53.64¢ beyond
the EV (48.23¢) of a $1 ticket
and has overall probability of winning:
.127
= about 1 in 7.9 (mostly from the $2 prizes)
Thanks for coming today!
“May the odds be ever in your favor!”
Professor Lesser
http://www.math.utep.edu/Faculty/lesser/lottery.html