Notesale: Turn your study into money

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Ryan  Passaro  

 

 
 

Math  Fair  2016  
Commack  Middle  School  
 

 

  

Abstract
 
In  my  project,  I  will  explain  the  basic  terms  and  rules  of  Texas  Hold‘em
...
 There  decisions  will  be  based  
upon  principles  of  probability,  combinatorics,  expected  value  and  pot  odds
...
 In  1967,  after  the  card  game  
spread  throughout  Texas,  it  was  introduced  to  Las  Vegas  by  a  group  of  people
...
 Addington  
said  he  first  saw  the  game  in  1955
...
 Van  De  Kamp
...
 
         Crandell  Addington  is  an  entrepreneur  and  a  famous  poker  player,  known  for  
being  one  of  the  founders  of  the  WSoP  (World  Series  of  Poker)  and  a  member  of  the  
Poker  Hall  of  Fame
...
 He  has  won  the  WSOP  twice  and  was  an  inductee  in  the  Poker  Hall  of  Fame
...
 In  
1979,  he  wrote  his  first  book  on  poker  strategy  and  wrote  seven  more,  one  of  his  
last  ones  on  online  poker
...
 He  was  
a  professional  poker  player  who  was  inducted  into  the  Hall  of  Fame  in  1992
...
 
 

Objective
 
Cards  are  dealt  randomly  a  total  of  two  cards  to  each  player  and  five  
community  cards
...
 The  objective  of  Texas  
Hold  ‘em  is  to  win  as  many  hands  as  a  player  can  win  by  mathematically  making  
better  decisions
...
 Each  player  can  attempt  to  control  the  amount  of  money  in  the  pot  
better  if  they  can  use  mathematics  in  their  strategy
...
 
Bluffing  is  a  technique  not  previously  mentioned  and  often  used  in  Texas  Hold  ‘em
...
 An  example  would  
be  to  raise  even  if  you  don’t  have  a  good  hand  to  try  and  convince  the  other  players  
your  hand  is  better  than  it  actually  is
...
 Television  and  internet  
were  two  reasons  why  at  the  turn  of  the  century  Texas  Hold  ‘em’s  popularity  
skyrocketed
...
 Texas  Hold  ‘em  was  also  in  many  
movies  such  as;  Rounders  (1998),  Casino  Royale  (2006),  and  Shark  out  of  Water  
(2008),  just  to  name  a  few
...
 This  debate  is  greater  explained  towards  
the  end  of  my  introduction
...
 These  are  
bets  that  two  people  must  put  in  during  the  pre-­‐flop
...
 In  some  cases,  the  game  is  also  played  with  antes
...
 The  difference  of  this  and  the  blind  bets  is  that  the  big/small  
blinds  must  put  in  their  bet  before  folding
...
   After  one  round  of  betting  is  done,  the  next  betting  round  will  
start    by  the  person  in    the  small  blind
...
 The  big  blinds’  bet  is  equivalent  to  the  other  player’s  antes
...
 During  the  “head  to  head”  time,  
the  player  in  the  dealer  position  is  the  small  blind
...
 
 

Play of the Hand

The  game  begins  with  two  cards  dealt  face-­‐down  to  each  player
...
 This  time  is  called  the  pre-­‐flop
...
 Next,  the  first  the  community  cards  are  dealt
...
 Any  player  may  use  these  cards
...
 Another  betting  round  
then  occurs
...
 The  turn  is  when  one  
more  community  card  is  dealt
...
 Finally  the  last  
community  card  is  dealt  called  the  “river”  or  “fifth  street”  with  the  last  betting  
round
...
 Every  time  a  new  community  card  is  
dealt,  a  card  is  ‘burned’
...


Strategy
Most  poker  professionals  recommend  an  aggressive  approach  to  a  game
...
 Mostly  all  poker  authors  agree  that  your  style  of  
play  depends  on  where  your  position  is  around  the  table
...
 
In  conclusion,  people  who  have  earlier  positions  usually  stay  in  less  rounds
...
 Famous  
Hold  ‘em  player  Phil  Hellmuth  once  said,  “If  it  weren’t  for  luck,  I’d  win  every  time
...
 On  the  flip  
side,  many  other  people  believe  that  no  matter  how  much  skill  is  involved  you  
cannot  predict  what  will  happen  next,  nor  control  the  future
...
 It  can  
be  calculated  by:  
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =

#  𝑜𝑓  𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒  𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
 
#  𝑜𝑓  𝑡𝑜𝑡𝑎𝑙  𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

 
For  example,  to  find  the  probability  of  picking  a  prime  number  out  of  a  group  
of  numbers  from  1-­‐12,  this  would  be  the  equation:  
 

#"of"favorable"outcomes:"in"this"case,"
any"prime"number"between"1712"
(2,"3,"5,"7,"11)"
#"of"total"possible"outcomes:"in"
this"case,"any"number"from"1712"
So  the  probability  of  picking  a  prime  number  out  of  the  numbers  1-­‐12  inclusive  
would  be  5  out  of  12,  or  41
...
 
 
 
 
 
 
 
 
 
 
 
 
 

2
 
5

Expected Value

  
What  is  expected  value?  Expected  value  is  the  predicted  value  of  a  variable
...
 Let’s  plug  in  
that  information  into  our  equation;  

 
𝐸𝑉 = %  𝑜𝑓  𝑤𝑖𝑛𝑛𝑖𝑛𝑔 10 + (%  𝑜𝑓  𝑙𝑜𝑠𝑖𝑛𝑔)(3)  

Now,  as  you  can  see,  we  need  to  calculate  the  percentage  of  losing  and  the  
percentage  of  losing;  

 
 
 
 
 
Probability*
Probability*
Probability*
Probability*
 
of*rolling*
of*rolling*
of*rolling*
of*rolling*
 
Probability*
double*
double*
double*
double*sixes*
 
of*rolling*
ones*
twos*
fours*
 
doubles*
 
Probability*
Probability*
 
of*rolling*
of*rolling*
 
double*
double*fives*
 
threes*
 
 
 
!
So  the  odds  of  rolling  doubles  are  !
...
 Let’s  plug  that  in
...
 
6
6
6
6
6
 If  the  expected  value  is  positive  you  should  play  because  even  though  you  can  lose  
𝐸𝑉 =

sometimes,  you  will  make  money  in  the  long  run
...
 So  in  this  case,  you  should  not  play
...
 In  a  jar,  there  are  11  blue  dice  and  16  red  dice
...
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Basic Permutation and Combination Skills
 

 

What  is  a  permutation?  A  permutation  is  a  way  that  a  set  of  numbers  can  be  
arranged  when  order  matters
...
 It  means  every  number  from  itself  to  1  is  multiplied
...
   
 
Example:    
How  many  different  ways  can  5  books,  (Math,  English,  Spanish,  
Chemistry,  and  Physics)  be  arranged  on  a  shelf  in  groups  of  4?    
 

Total&number&of&books&

 
 
This  can  be  written  as;  

!!

!! !∙!∙!∙!∙!

= =

(!!!)! !!

!

 
 
 
 
 
Number&of&books&in&a&group&  
 
 

=5  ∙ 4 ∙ 3 ∙ 2=120  

 
So  there  are  120  ways  to  arrange  four  books  on  a  shelf
...
 
 
What  is  a  combination?  A  combination  is  a  way  that  a  set  of  numbers  can  be  
! 𝑃
!

=

arranged  when  order  doesn’t  matter
...
 
 
 
 
 
 

Calculating A Hand
 
Let’s  calculate  the  chance  of  getting  a  Full  House:  

A  full  house  is  a  three  of  a  kind  and  a  pair  within  your  five  cards
...
 So  we  have  13  choices  and  we  are  choosing  one  of  them,  because  
we  need  a  card  number  for  the  first  three  cards
...
 We  also  have  four  
suits  and  we  are  choosing  3  or  have  3  choices  for
...
 We  
multiply  these  numbers  together
...
 We  are  choosing  
between  12  different  card  numbers  now  because  we  cannot  have  a  four  of  a  kind
...
 We  also  have  4  suits  and  are  choosing  2
...
 
Now,  we  multiply  them  all;  
 

!" 𝐶!

∙

! 𝐶!

∙ !" 𝐶! ∙

! 𝐶!

 

13!
4!
12!
4!
∙
∙
∙
 
1! 13 − 1 ! 3! 4 − 3 ! 1! 12 − 11 ! 2! 4 − 2 !
13! 4! 12! 4!
∙ ∙
∙  
12! 3! 11! 4
13 ∙ 4 ∙ 12 ∙ 6 = 3,744  
Next,  we  have  to  divide  this  number  by  the  total  possible  hands  you  could  
have
...
 2,580,960  is  calculated  by  there  are  52  cards  in  a  
deck,  and  5  cards  in  a  hand
...
 
 

!" 𝐶!

= 2,598,960  

3,744
1
= 694  
2,598,960
6
!

So  the  odds  of  getting  a  full  house  are  1  in  694!
...
 This  is  the  skill  
portion  of  the  game
...
 Should  I  call,  raise,  or  fold?  
 
The  only  possible  hand  that  your  opponent  could  get  that  would  beat  you  would  be  
the  full  house
...
   
 
 
 
 
 
My%2%cards%+%
opponents%2%
 
cards%+%4%flop%
There%is%a%
cards%are%
 
King%of%
known
...
 Should  you  call?  You  
should  only  call  if  your  odds  of  winning  are  greater  than  your  pot  odds
...
 Your  pot  odds  are  your  contribution  to  the  pot  to  play  as  a  fraction  of  the  
total
...
 Raising  has  two  purposes
...
   
!
 
If  I  put  myself  in  my  opponents  shoes,  my  odds  of  winning  is  !!,  what  raise  
would  I  have  to  call,  to  make  my  pot  odds  greater  than  my  implied  odds?  (Only  then  
would  I  play)
...
11𝑥 > 220 + 2𝑥  
220 + 2𝑥 11
9𝑥 > 220  
4
𝑥 > 24  
9
So  if  you  bet  anything  above  $25,  he  should  fold
...
 

 

 
 
 
 
 
 

Conclusion
               
                 I  am  very  interested  in  this  topic  and  that  is  why  I  chose  to  study  the  
mathematics  behind  the  game  of  Texas  Hold’em;  my  family  and  I  play  often,  so  I  
hope  to  use  the  skills  I’ve  learned  in  the  future
...
 Expected  value  calculations  can  help  me  decide  when  
hands  should  be  played  or  folded  
...
 
           Also,  I  realized  how  math  is  used  in  our  everyday  lives:  lottery,  insurance  
policies,  or  any  risk  analysis  involves  similar  expected  value  calculations
...
 If  I  had  more  time  to  continue  my  research  I  would  have  situations  with  
more  than  two  people,  situations  on  the  turn  or  different  poker  card  games
...
com
...
”  Poker  Terms  and  Definitions
...
p
...
 Web
...
 2015
...
pokerstars
...
 
I  used  this  website  to  explains  terms  of  Texas  Hold’em
...
 “Permutations  and  Combinations
...
 N
...
,  n
...
   
Web
...
 2015
...
regentsprep
...
htm
...
 
 
Young,  Wesley  R
...
”  HowStuffWorks
...
com,  07  Apr
...
 Web
...
 2015
...
hotstuffworks
...
htm
...
   
 
Cotter,  Neil  E
...
”  (n
...
):  n
...
 1  Feb
...
 
Web
...
 2015
...
ecu
...
edu/eceCTools/Probability/Combinatorics/ProbCombEx15
...
 
I  used  this  website  to  explain  calculating  the  odds  of  getting  a  hand  of  poker
...
 Doyle  Brunson's  Super  System:  A  Course  in  
Power  Poker
...
,  2002
...
   
I  used  this  book  to  get  a  better  understanding  of  poker  strategy

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