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Florida Lotto Probability

Topic closed. 8 replies. Last post 3 years ago by RJOh.

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New Member
Gainesville, FL
United States
Member #121739
January 16, 2012
8 Posts
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Posted: January 22, 2012, 7:47 pm - IP Logged

Florida Lotto is a 6 ball, 53 number Lottery.
It's Odds are:

Hits = 1     1 in 8.8                                         
Hits = 2     1 in 91.9                                       
Hits = 3     1 in 1171.3                                     
Hits = 4     1 in 19521.7                                     
Hits = 5     1 in 478280.8                                   
Hits = 6     1 in 22957480

There are around 104 drawings per year; so you could expect to Hit 1 ball on your quick pick about once a month ... 2 about once a year ... 3 about once a decade ... 4 about once a generation ... 5 once in a lifetime and 6 in the midst of the next ice age.

    RJOh's avatar - chipmunk
    mid-Ohio
    United States
    Member #9
    March 24, 2001
    18020 Posts
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    Posted: January 23, 2012, 11:48 pm - IP Logged

    Where did you come up with those odds calculations?

     possible combos of 6/53 numbers = 22957480
     MATCHES     ODDS               
      6/6      1 : 22957480           
      5/6      1 : 81410             
      4/6      1 : 1416               
      3/6      1 : 71                 
      2/6      1 : 9                 
      1/6      1 : 2                 

    My odds calculations show you can expect to match one on every other QP or 1:2 or get a free ticket once a month or 1:9.

    * you don't need more tickets, just the right ticket * 
    * your best chance at winning a lottery jackpot is to buy a ticket * 
         Wink 

      Avatar
      New Member
      Gainesville, FL
      United States
      Member #121739
      January 16, 2012
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      Posted: January 24, 2012, 6:07 pm - IP Logged

      C(53,Hits)/C(6,Hits)
      Essentially, Make 2 lists ... one with all possible combinations using 53 numbers, in the other, all possible combinations using 6 numbers.  Then divide the list counts.

      You get a slightly better result if you do: 1/(1/53 + 1/52 + 1/51 + 1/50 + 1/49 + 1/48)  = 1 in 8.4, but I prefer the picking entries in a great big list method.

      HitsOddsCombos in 53Combos in 6
      Hits = 11 in 8.8536
      Hits = 21 in 91.9137815
      Hits = 31 in 1171.32342620
      Hits = 41 in 19521.729282515
      Hits = 51 in 478280.828696856
      Hits = 61 in 22957480229574801
        Avatar
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        Gainesville, FL
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        Member #121739
        January 16, 2012
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        Posted: January 24, 2012, 6:22 pm - IP Logged

        * THat which happens most *
        * is most likely to happen again *
         

        is unlikely in a lottery.

          Avatar
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          Gainesville, FL
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          January 16, 2012
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          Posted: January 24, 2012, 7:00 pm - IP Logged

          I don't know if you're a coder; but these are my visual basic routines -- p is permutations, c is combinations.

          ' P(n,k) = n! / (n-k)!
          Public Function p(ByVal n As Double, ByVal k As Double) As Double
              Dim i As Integer
             
              p = 1
             
              For i = ((n - k) + 1) To n
                  p = i * p
              Next
          End Function

          ' C(n,k) = P(n,k) / k!
          Public Function c(ByVal n As Double, ByVal k As Double) As Double
              Dim i As Integer
             
              c = p(n, k)
             
              For i = 2 To k
                  c = c / i
              Next i
          End Function

          I'd really like to know if I've got any of this wrong.

            RJOh's avatar - chipmunk
            mid-Ohio
            United States
            Member #9
            March 24, 2001
            18020 Posts
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            Posted: January 24, 2012, 7:21 pm - IP Logged

            I don't know if you're a coder; but these are my visual basic routines -- p is permutations, c is combinations.

            ' P(n,k) = n! / (n-k)!
            Public Function p(ByVal n As Double, ByVal k As Double) As Double
                Dim i As Integer
               
                p = 1
               
                For i = ((n - k) + 1) To n
                    p = i * p
                Next
            End Function

            ' C(n,k) = P(n,k) / k!
            Public Function c(ByVal n As Double, ByVal k As Double) As Double
                Dim i As Integer
               
                c = p(n, k)
               
                For i = 2 To k
                    c = c / i
                Next i
            End Function

            I'd really like to know if I've got any of this wrong.

             nCr=n!/[(n-r)!*r!]

            I used the above formula to get my results which are the same posted on state websites that I've visited so I assume they are correct.

            *Permutations aren't the same as a combinations.  Permutations include all the different possible arrangements of numbers in a combination.

            * you don't need more tickets, just the right ticket * 
            * your best chance at winning a lottery jackpot is to buy a ticket * 
                 Wink 

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              New Member
              Gainesville, FL
              United States
              Member #121739
              January 16, 2012
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              Posted: January 24, 2012, 8:15 pm - IP Logged

              Ok.  I've put up my thinking; but I also ran a test.  I took the first 100 Lotteries in the Florida Lotto history list and ran them against each other so that each Lottery took the other 99 as Quick Picks.  The results confirm your calculations.  Where did I go wrong?

              Quick Pick Results
              HitsCountPercentProbability
              1404440.851 in 2.4
              2117011.821 in 8.5
              31181.191 in 83.9
              4100.11 in 990.0
                Avatar
                New Member
                Gainesville, FL
                United States
                Member #121739
                January 16, 2012
                8 Posts
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                Posted: January 25, 2012, 7:03 am - IP Logged

                Thanks RJOh!
                My error is that I thought the probability formula is WaysToWin/TotalCombinations.  The correct formula is (WaysToWin*WaysToLose)/TotalCombinations.  The Permutations and Combinations code I put up are correct and this is the corrected table, now just like RJOh:

                Odds

                HitsPicks to HitWays to WinWays to LoseTotal Combinations
                02.111073757322957480
                12.56153393922957480
                28.61517836522957480
                370.8201621522957480
                41415.815108122957480
                581409.564722957480
                6229574801122957480
                  RJOh's avatar - chipmunk
                  mid-Ohio
                  United States
                  Member #9
                  March 24, 2001
                  18020 Posts
                  Offline
                  Posted: January 25, 2012, 5:06 pm - IP Logged

                  Thanks RJOh!
                  My error is that I thought the probability formula is WaysToWin/TotalCombinations.  The correct formula is (WaysToWin*WaysToLose)/TotalCombinations.  The Permutations and Combinations code I put up are correct and this is the corrected table, now just like RJOh:

                  Odds

                  HitsPicks to HitWays to WinWays to LoseTotal Combinations
                  02.111073757322957480
                  12.56153393922957480
                  28.61517836522957480
                  370.8201621522957480
                  41415.815108122957480
                  581409.564722957480
                  6229574801122957480

                  That's pretty close.  These are some full printouts of the program I wrote some years ago using GWBasic.  I can use different variables to fit the game that I'm playing or the answers that I want.

                   combination size             6
                   basic pool size              53
                   (B) bonus numbers            none
                   smallest match               0
                   tickets or chances per draw  1
                   possible combos of 6/53 numbers = 22957480
                   MATCHES     ODDS                WINNING COMBOS      EXPECTED WINNERS
                    6/6      1 : 22957480           1                      0.00
                    5/6      1 : 81410              282                    0.00
                    4/6      1 : 1416               16215                  0.00
                    3/6      1 : 71                 324300                 0.01
                    2/6      1 : 9                  2675475                0.12
                    1/6      1 : 2                  9203634                0.40
                    0/6      1 : 2                  10737573               0.47
                   ______________________________________________________________________________
                   overall odds are 1 : 1                  1.0 total expected winners
                   22957480 winning combos = 100 % of possible

                   combination size             5
                   basic pool size              59
                   (B) Bonus pool size          35
                   smallest match no (B) number 3
                   largest match with bonus     5
                   smallest match with bonus    0
                   tickets or chances per draw  1
                   possible combos of 5/59 + 1/35 numbers = 175223510
                   MATCHES     ODDS                WINNING COMBOS      EXPECTED WINNERS
                    5/5+B    1 : 175223510          1                      0.00
                    5/5+0    1 : 5153633            34                     0.00
                    4/5+B    1 : 648976             270                    0.00
                    4/5+0    1 : 19088              9180                   0.00
                    3/5+B    1 : 12245              14310                  0.00
                    3/5+0    1 : 360                486540                 0.00
                    2/5+B    1 : 706                248040                 0.00
                    1/5+B    1 : 111                1581255                0.01
                    0/5+B    1 : 55                 3162510                0.02
                   ______________________________________________________________________________
                   overall odds are 1 : 31.8               0.0 total expected winners
                   5502140 winning combos = 3.14 % of possible combos

                   combination size             6
                   basic pool size              53
                   (B) bonus numbers            none
                   smallest match               3
                   tickets or chances per draw  1
                   possible combos of 6/53 numbers = 22957480
                   MATCHES     ODDS                WINNING COMBOS      EXPECTED WINNERS
                    6/6      1 : 22957480           1                      0.00
                    5/6      1 : 81410              282                    0.00
                    4/6      1 : 1416               16215                  0.00
                    3/6      1 : 71                 324300                 0.01
                   ______________________________________________________________________________
                   overall odds are 1 : 67.3               0.0 total expected winners
                   340798 winning combos = 1.48 % of possible combos

                  * you don't need more tickets, just the right ticket * 
                  * your best chance at winning a lottery jackpot is to buy a ticket * 
                       Wink