welcome to…

“A Lotto Learning: Lottery Literacy”

with Dr. Lawrence (Larry) LesserProfessor, 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

• 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 $ goeshttp://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 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 choose k” = ,

where n! is product of first n positive integers (so 4! = 4x3x2x1 = 24)

• Lotto Texas (n = 54, k = 6): = 25,827,165 (can also verify via the hypergeometric distribution)

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?• 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:• 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• 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 StoryTeacher 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• 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: ($100) + ($10) = $28

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:

Pr(no repeats) = = 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:

• Note: # of combos increases fast! To buy all 6-number combos from 1-10 takes $210!

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

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 poolingwww.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:

• TV/radio/magazine interviews

• award-winning YouTube video

• Lottery Literacy webpage http://www.math.utep.edu/Faculty/lesser/lottery.html

• and a SONG!

J. of Statistics Education March 2013

Mathematics Teacher Sept. 2012

Statistics Teacher Network Winter 2004

Texas Mathematics Teacher Fall 2003

Spreadsheet User Nov. 1997

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 thisAt 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 thisAt 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 – = 1 – ≈ .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 6-ball 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….

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)

# 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

Thanks for coming today!

“May the odds be ever in your favor!”

Professor Lesser http://www.math.utep.edu/Faculty/lesser/lottery.html