There's a saying -- "Mathematicians don't gamble." Whilst this may be true of soccer or horse racing betting,  this week, I'll be looking at Keno, Bingo, and various lotteries. All of these can be analyzed by looking at the Urn problem  An urn has 2 red balls, and 6 green balls.  Four balls are drawn at random.  What are the odds of drawing both red balls?  Let red = # of red balls, green = # of green balls, draws = # of balls drawn, and need = # of red balls needed.  The Factorial (!) is a product (7! = 1*2*3*4*5*6*7 = 5040).  Note that 0! = 1.  The urn problem uses the following formula:

In Keno, you have a grid of 80 numbers.  You choose 20 of them.  Then, 20 numbers are picked at random.  If enough numbers match yours, you win a prize.  This is equivalent to 20 red balls, 60 green, 20 draws, and a variable need.  If you need to match 10 numbers, you'll win approximately once out of every 254 games.  You can see a list of Keno odds here, or you can use the formula to calculate them yourself.

In Bingo, you have a square of 24 numbers, with a free space at the center.  There are 75 total numbers.    If you are going for a full card, the problem is equivalent to 24 red balls, 51 green balls, a variable number of draws, and 24 red balls needed.  Things start getting interesting around 57 balls drawn, where there is a 1 in 3427 chance of winning.  After that; 58 draws = 1/2009, 59 draws = 1/1192, 60 draws = 1/715, 61 draws = 1/434, 62 draws = 1/266, 63 draws = 1/165.  By now, in a room of 300 people, there is a (1 - (164/165)³⁰⁰) = 84% chance that someone has already called Bingo.  There, they stop at 48 draws, so there is a 1 in 799399 chance of winning.

Before going into a more complicated Bingo problem, consider rolling three fair 6-sided dice.  What are the odds of getting at least one six?  With each die, there is a (1 - 1/6) = 5/6 chance that a six will not come up.  For uncorrelated probabilities, these events can be multiplied:  (1 - 1/6)³ = 58% chance that no sixes will come up.  There is a 1 - (1 - 1/6)³ = 42% chance of getting at least one six.

In Bingo, consider getting five in a row.  For this, there are 8 ways of getting a pure five in a row, and 4 ways of going through the center.  For pure five, the urn problem 5 red, 70 green, variable draws, need of 5 applies ==> Urn[5, 70, X, 5].  For using the center, the urn problem 4 red, 71 green, variable draws, need of 4 applies ==> Urn[4, 71, X, 4].  Getting a line in one direction isn't correlated to another line, so we can use the same trick as the dice example.

Odds of getting five in a row in Bingo = 1  -  ( 1 - Urn[5, 70, X, 5] )⁸  *  (1 - Urn[4, 71, X, 4] )⁴ .   The odds of someone getting five in a row in ten draws = 1/1238, 11 draws = 1/770, 12 draws = 1/501, 13 draws = 1/339, 14 draws = 1/237, 15 draws = 1/170.  Someone has probably called Bingo by this point.  After 15 draws, one out of every eight people is close to winning (Use 1  - ( 1 - Urn[5, 70, X, 4] )⁸  *  (1 - Urn[4, 71, X, 3] )⁴ ).  Most Bingo players have multiple cards, and play multiple times per night.  Almost all players will thus come close to winning several times in the course of a night.  Hence the appeal.

The Lottery varies from state to state, and country to country.  If you don't see your locale's lottery game listed here, write to me and I'll add it.  Calculating Lottery odds needs only the Binomial, where n = # of numbers, and k = # of picks.  Gerry Quinn has put together a Lotto Odds Calculator you might like to try.

Obviously, there is a lot of gambling on this planet.  You might be unfamiliar with Lottery Wheels.  A problem in Combinatorics: Buy n tickets for a lottery so that at least one of the tickets will match three numbers.  The typical payoff for matching 3 numbers is $3, I believe. The numbers in parentheses are the smallest known Lottery Wheels.  For the Australia Lotto (6 out of 45), you could guarantee yourself a win of $3.00 just by spending $154.00 on tickets.  I'll beat that!  Just send me $154.00, and I'll send you $10.00.  For 44 or 45 picks, you'll need the Steiner S(3, 6, 22) system.  That's three matches out of six given 22 numbers.  You can obtain various systems of this sort from the La Jolla Covering Repository by Dan Gordon.  There, I used v=22, k=6, t=3, and got the following:

The 77 A Tickets = [1 2 3 4 5 6], [1 2 7 8 9 10], [1 2 11 12 13 14], [1 2 15 16 17 18], [1 2 19 20 21 22], [1 3 7 11 15 19], [1 3 8 12 16 20], [1 3 9 13 17 21], [1 3 10 14 18 22], [1 4 7 12 17 22], [1 4 8 11 18 21], [1 4 9 14 15 20], [1 4 10 13 16 19], [1 5 7 13 18 20], [1 5 8 14 17 19], [1 5 9 11 16 22], [1 5 10 12 15 21], [1 6 7 14 16 21], [1 6 8 13 15 22], [1 6 9 12 18 19], [1 6 10 11 17 20], [2 3 7 12 18 21], [2 3 8 11 17 22], [2 3 9 14 16 19], [2 3 10 13 15 20], [2 4 7 11 16 20], [2 4 8 12 15 19], [2 4 9 13 18 22], [2 4 10 14 17 21], [2 5 7 14 15 22], [2 5 8 13 16 21], [2 5 9 12 17 20], [2 5 10 11 18 19], [2 6 7 13 17 19], [2 6 8 14 18 20], [2 6 9 11 15 21], [2 6 10 12 16 22], [3 4 7 8 13 14], [3 4 9 10 11 12], [3 4 15 16 21 22], [3 4 17 18 19 20], [3 5 7 10 16 17], [3 5 8 9 15 18], [3 5 11 14 20 21], [3 5 12 13 19 22], [3 6 7 9 20 22], [3 6 8 10 19 21], [3 6 11 13 16 18], [3 6 12 14 15 17], [4 5 7 9 19 21], [4 5 8 10 20 22], [4 5 11 13 15 17], [4 5 12 14 16 18], [4 6 7 10 15 18], [4 6 8 9 16 17], [4 6 11 14 19 22], [4 6 12 13 20 21], [5 6 7 8 11 12], [5 6 9 10 13 14], [5 6 15 16 19 20], [5 6 17 18 21 22], [7 8 15 17 20 21], [7 8 16 18 19 22], [7 9 11 14 17 18], [7 9 12 13 15 16], [7 10 11 13 21 22], [7 10 12 14 19 20], [8 9 11 13 19 20], [8 9 12 14 21 22], [8 10 11 14 15 16], [8 10 12 13 17 18], [9 10 15 17 19 22], [9 10 16 18 20 21], [11 12 15 18 20 22], [11 12 16 17 19 21], [13 14 15 18 19 21], [13 14 16 17 20 22].

For the 77 B Tickets, just add 22 to every number in every A Ticket.  Now you have a batch of 154 tickets.

If the Lottery Numbers are 1, 4, 16, 20, 23, 35; then the above system will give you four 3-matches.  It still loses money.  For a bit more randomness, you can swap numbers in your list.  For example, change every 1 in the full list to 27, and change every 27 to 1.
For (6  44), use the above 154 tickets.
For (6  45), use the above 154 tickets.
For (6  46), add six tickets [1 2 3 4 45 46], [5 6 7 8 45 46], [9 10 11 12 45 46], [13 14 15 16 45 46], [17, 18, 19, 20, 45, 46], [X, X, X, 21, 22, 45].
For (6  47), add seven tickets [1 2 3 45 46 47], [4 5 6 45 46 47], ... , [19 20 21 45 46 47].
For (6  48), add eleven tickets [1 2 45 46 47 48], [3 4 45 46 47 48], [5 6 45 46 47 48], ... , [21 22 45 46 47 48].
For (6  49), add twenty tickets [6 45-49], [7 45-49], ... , [22 45-49], [1 2 45-48], [3 4 45-48], [1-5 49].
For (6  50), add thirty four tickets ([1 2 45-48], [1 2 47-50], [1 2 45 46 49 50]), ([3 4 45-48], [3 4 47-50], [3 4 45 46 49 50]), ... , ([21 22 45-48], [21 22 47-50], [21 22 45 46 49 50]), [45-50].
For (6  51), see the CRC Handbook of Combinatorial Design.  That is where I learned about Lottery Wheels.  If you can improve any of them, let me know.
For (6  36), start with the 48 C tickets.  Add D tickets -- the same group, with 18 added to everything.  C = [1 3 5 7 8 10], [2 3 9 10 11 12], [4 6 7 10 11 13], [3 4 5 6 9 14], [2 3 6 8 13 14], [1 4 10 12 13 14], [2 5 6 7 11 15], [3 4 8 11 12 15], [1 3 7 9 13 15], [4 5 8 10 13 15], [2 5 9 10 13 15], [1 2 5 12 14 15], [1 2 6 9 10 16], [4 5 7 9 12 16], [2 7 8 12 13 16], [1 3 7 11 14 16], [5 8 10 11 14 16], [1 6 11 12 14 16], [1 4 6 8 15 16], [9 11 13 14 15 16], [1 2 3 4 7 17], [2 5 8 9 10 17], [6 7 8 9 11 17], [2 4 9 10 13 17], [3 5 11 12 13 17], [1 8 9 12 14 17], [2 4 5 8 15 17], [4 8 9 13 15 17], [3 6 10 14 15 17], [1 5 6 13 16 17], [2 4 11 14 16 17], [7 10 12 15 16 17], [1 4 5 9 11 18], [1 3 6 7 12 18], [5 6 8 10 12 18], [1 2 8 11 13 18], [2 7 9 10 14 18], [6 7 11 12 14 18], [6 9 12 13 15 18], [4 7 8 14 15 18], [3 6 7 11 16 18], [3 4 10 13 16 18], [1 3 12 14 16 18], [2 3 5 15 16 18], [2 4 6 12 17 18], [5 7 13 14 17 18], [1 10 11 15 17 18], [3 8 9 16 17 18].
For (6  37) to (6  43), start with the C and D tickets, and follow the methods for adding to A and B tickets.

Alfons Lievens of Belgium has found a 42 numbers solution with 120 tickets. Here it is.

Siegbert Steinlechner informed me of a Lotto 6/49 wheel with 163 tickets: [1,2,3,4,5,10], [1,2,9,10,11,26], [1,2,9,10,14,23], [1,2,10,12,16,20], [1,3,14,20,23,26], [1,4,9,12,15,27], [1,4,12,14,18,23], [1,5,11,14,16,27], [1,5,11,15,16,23], [1,6,7,10,18,19], [1,6,8,17,21,27], [1,6,13,15,18,25], [1,7,8,15,18,24], [1,7,13,17,22,27], [1,8,10,17,22,25], [1,13,14,21,23,24], [1,15,18,19,21,22], [1,17,19,24,25,27], [2,3,15,20,26,27], [2,4,9,12,14,27], [2,4,9,12,15,23], [2,5,11,14,16,23], [2,6,8,14,18,21], [2,6,13,17,23,25], [2,6,15,22,24,27], [2,7,8,17,23,24], [2,7,13,14,18,22], [2,7,15,21,25,27], [2,8,13,15,19,27], [2,10,15,17,18,27], [2,14,18,19,24,25], [2,17,19,21,22,23], [3,4,5,13,21,24], [3,4,5,14,15,17], [3,4,5,18,23,27], [3,4,13,16,17,26], [3,5,10,11,12,13], [3,6,7,8,20,26], [3,6,7,11,12,19], [3,6,9,16,22,24], [3,7,9,16,21,25], [3,8,9,13,16,19], [3,8,11,12,22,25], [3,9,10,16,17,18], [3,9,11,12,17,21], [3,11,12,18,20,24], [3,19,20,22,25,26], [4,5,9,13,18,20], [4,6,7,16,19,26], [4,6,9,12,19,25], [4,6,11,20,22,24], [4,7,8,9,12,22], [4,7,11,20,21,25], [4,8,11,13,19,20], [4,8,16,22,25,26], [4,10,11,17,18,20], [4,10,12,16,24,26], [4,11,16,18,21,26], [5,6,7,9,19,20], [5,6,8,11,16,25], [5,6,12,22,24,26], [5,7,11,16,19,22], [5,7,12,21,25,26], [5,8,9,20,22,25], [5,8,12,13,19,26], [5,9,10,20,21,26], [5,9,16,17,20,24], [5,10,12,17,18,26], [6,7,21,22,23,24], [6,8,10,15,21,23], [6,8,14,17,19,22], [6,10,13,14,25,27], [7,8,10,14,24,27], [7,10,13,15,22,23], [7,14,15,17,23,25], [8,13,19,21,24,25], [8,18,19,23,25,27], [9,11,13,21,24,26], [9,11,14,15,17,26], [9,11,18,23,26,27], [10,13,17,18,21,24], [10,14,19,21,22,27], [10,15,19,23,24,25], [12,13,16,20,21,24], [12,14,15,16,17,20], [12,16,18,20,23,27], [28,29,30,31,32,49], [28,29,33,38,44,47], [28,29,34,37,43,48], [28,29,35,40,42,45], [28,29,36,39,41,46], [28,30,33,37,42,46], [28,30,34,38,41,45], [28,30,35,39,44,48], [28,30,36,40,43,47], [28,31,33,40,41,48], [28,31,34,39,42,47], [28,31,35,38,43,46], [28,31,36,37,44,45], [28,32,33,39,43,45], [28,32,34,40,44,46], [28,32,35,37,41,47], [28,32,36,38,42,48], [28,33,34,35,36,49], [28,37,38,39,40,49], [28,41,42,43,44,49], [28,45,46,47,48,49], [29,30,33,34,39,40], [29,30,35,36,37,38], [29,30,41,42,47,48], [29,30,43,44,45,46], [29,31,33,36,42,43], [29,31,34,35,41,44], [29,31,37,40,46,47], [29,31,38,39,45,48], [29,32,33,35,46,48], [29,32,34,36,45,47], [29,32,37,39,42,44], [29,32,38,40,41,43], [29,33,37,41,45,49], [29,34,38,42,46,49], [29,35,39,43,47,49], [29,36,40,44,48,49], [30,31,33,35,45,47], [30,31,34,36,46,48], [30,31,37,39,41,43], [30,31,38,40,42,44], [30,32,33,36,41,44], [30,32,34,35,42,43], [30,32,37,40,45,48], [30,32,38,39,46,47], [30,33,38,43,48,49], [30,34,37,44,47,49], [30,35,40,41,46,49], [30,36,39,42,45,49], [31,32,33,34,37,38], [31,32,35,36,39,40], [31,32,41,42,45,46], [31,32,43,44,47,48], [31,33,39,44,46,49], [31,34,40,43,45,49], [31,35,37,42,48,49], [31,36,38,41,47,49], [32,33,40,42,47,49], [32,34,39,41,48,49], [32,35,38,44,45,49], [32,36,37,43,46,49], [33,34,41,43,46,47], [33,34,42,44,45,48], [33,35,37,40,43,44], [33,35,38,39,41,42], [33,36,37,39,47,48], [33,36,38,40,45,46], [34,35,37,39,45,46], [34,35,38,40,47,48], [34,36,37,40,41,42], [34,36,38,39,43,44], [35,36,41,43,45,48], [35,36,42,44,46,47], [37,38,41,44,46,48], [37,38,42,43,45,47], [39,40,41,44,45,47], [39,40,42,43,46,48]

The trouble I see with most gambling these days is a lack of creativity.  Where are the new games?  The last time I played poker, the series of dealers introduced to dozens of variations (I lost $7 in three hours).  Why can't the big gambling houses and online poker sites be as creative as poker players?  They could do something with the 4 fair 60-sided dice, with 12960000 possible outcomes.

How about the game of Letto?  This would be like Lotto, but with letters.  If you get all five letters correct in the right sequence, you win the jackpot.  There are 7893600 possible tickets.  An alphabetic series would have a sequence 12345.  ERDOS would have the sequence 24135.  If the sequence of your letters matches the chosen sequence, regardless of whether letters match, you would win $20 (1 in 120 chance).  If you match three letters in the wrong places, you win $5.  If you match a letter in the correct place, you win a free ticket.  If the winning ticket is an english word, PRISM gets a free ticket, RULES wins $5, and ENVOI wins $20, what is a possible winning ticket? Feel free to send me an analysis of Letto.  Are there any interesting schemes that can be used for it?  I'd also like to hear of other new gambling games.

Casanova was not the only one to study gambling.  The Chevalier De Mere was an avid gambler that asked Blaise Pascal for an answer to a probability question.  Pascal solved the question, and sent it to Pierre Fermat and Christian Huygens.  The science known as Probability Theory was born.