Lottery Math: Fantasy 5 and Cash 5 Odds and Payouts
"Cash 5," aka "Fantasy 5," aka "Pick 5" are very popular and simple fivenumber draw lottery games played in most states. Tickets cost $1 and usually consist of a single play. While games bearing these names are played in more than one state, they are not technically "multistate" lotteries such as Powerball or Mega Millions, since each state operates Fantasy 5 or Cash 5 independently.
The Fantasy 5 and Cash 5 ticket sales are not pooled by the states to create large jackpots as in Powerball and Mega Millions. Each state maintains its own pool of prize funds, holds its own drawings, and determines the prize amounts for the jackpot and lower prize levels. Though the drawings may be held on the same day, the winning numbers may be different. For example, on Dec 25, 2013, the winning Fantasy 5 numbers in California were 0107162223, while in Florida on they were 0824283436 on the same day.
Nevertheless, the structures of the games and the probabilities of winning various prizes are basically the same in all states that offer lotteries by the name of Fantasy 5 or Cash 5.
How to Play Fantasy 5 or Cash 5
Fantasy 5 and Cash 5 lottos have the same simple structure no matter where they're played; you select 5 distinct numbers from 1 to X. The value of X typically ranges from 31 to 41, but may be as high as 47.
For example, in California Fantasy 5 and Indiana Cash 5, X = 39, therefore you select five numbers from between 1 and 39. In Florida Fantasy 5 and Oklahoma Cash 5, X = 36, therefore you pick five numbers from between 1 and 36.
Odds of Winning Fantasy 5 or Cash 5
The lower the value of X, the fewer the number of lottery ticket combinations there are, which means a higher probability of winning. Conversely, the larger the value of X, the worse the odds are. To make up for the disparity in odds, games with a lower probability of winning usually offer larger cash prizes.
The probability of matching 5 out of 5 numbers and winning the jackpot or grand prize is given by the combinatorial expression
1/(X choose 5)
= 5! * (X5)! / X!
= 120/[X(X1)(X2)(X3)(X4)]
The probability of matching 4 out of 5 is
5*(X5)*5!*(X5)!/X!
= 600(X5)/[X(X1)(X2)(X3)(X4)]
The probability of matching 3 out of 5 is
5*(X5)*(X6)*5!*(X5)!/X!
= 600(X5)(X6)/[X(X1)(X2)(X3)(X4)]
The probability of matching 2 out of 5 is
5*(X5)*(X6)*(X7)*5!*(X5)!/(3*X!)
= 200(X5)(X6)(X7)/[X(X1)(X2)(X3)(X4)]
In these expressions above, replace X with range of the lottery number field. For example, to compute the odds of winning California Fantasy 5, you replace X with 39. This gives you the probabilities
Prob(5 of 5) = 1/575757 = 0.0000017368 = 1 in 575,757
Prob(4 of 5) = 170/575757 = 0.00029526 = 1 in 3,387
Prob(3 of 5) = 1870/191919 = 0.0097437 = 1 in 103
Prob(2 of 5) = 59840/575757= 0.10393 = 1 in 10
The table below gives the odds for various matches for values of X between 31 and 47.
X
 5/5 Odds
 4/5 Odds
 3/5 Odds
 2/5 Odds


31
 1 in 169,911
 1 in 1,307
 1 in 52
 1 in 7

32
 1 in 201,376
 1 in 1,492
 1 in 57
 1 in 7

33
 1 in 237,336
 1 in 1,695
 1 in 63
 1 in 7

34
 1 in 278,256
 1 in 1,919
 1 in 69
 1 in 8

35
 1 in 324,632
 1 in 2,164
 1 in 75
 1 in 8

36
 1 in 376,992
 1 in 2,432
 1 in 81
 1 in 8

37
 1 in 435,897
 1 in 2,724
 1 in 88
 1 in 9

38
 1 in 501,942
 1 in 3,042
 1 in 95
 1 in 9

39
 1 in 575,757
 1 in 3,387
 1 in 103
 1 in 10

40
 1 in 658,008
 1 in 3,760
 1 in 111
 1 in 10

41
 1 in 749,398
 1 in 4,163
 1 in 119
 1 in 10

42
 1 in 850,668
 1 in 4,598
 1 in 128
 1 in 11

43
 1 in 962,598
 1 in 5,066
 1 in 137
 1 in 11

44
 1 in 1,086,008
 1 in 5,569
 1 in 147
 1 in 12

45
 1 in 1,221,759
 1 in 6,109
 1 in 157
 1 in 12

46
 1 in 1,370,754
 1 in 6,687
 1 in 167
 1 in 13

47
 1 in 1,533,939
 1 in 7,304
 1 in 178
 1 in 13

All odds in this table are rounded to the nearest integer. Notice that adding several more balls to a game can drastically change the odds. From the table you can see that whenever you add 5 more balls to the total range of numbers, the odds of making a perfect 5/5 match become roughly twice as bad.
This often surprises people who assume a lottery game with 20% or 50% more numbers to choose from has 20% or 50% higher odds against winning. The odds against winning do not increase linearly or in proportion to the number of extra balls added. A small change can make a big difference mathematically.
Table of State Lottery Cash 5 and Fantasy 5 Games
Here is a table of various Cash 5style games for different states. Some states offer more than one type of this game with different "X" values. Many states do not offer this kind of game (or any lottery at all.) Gimme 5 is a tristate game, meaning the money is pooled by the three states that offer it in order to create a larger jackpot. This is because each of the three member states (New Hampshire, Vermont, and Maine) has a relatively small population.
State
 Name of Game
 Value of X
 Explanation


Arizona
 Fantasy 5
 41
 Players select five numbers from 1 to 41.

Arkanasas
 Natural State Jackpot
 39
 Players select five numbers from 1 to 39.

California
 Fantasy 5
 39
 Players select five numbers from 1 to 39.

Colorado
 Cash 5
 32
 Players select five numbers from 1 to 32.

Connecticut
 Cash 5
 35
 Players select five numbers from 1 to 35.

Florida
 Fantasy 5
 36
 Players select five numbers from 1 to 36.

Georgia
 Fantasy 5
 42
 Players select five numbers from 1 to 42.

Idaho
 Weekly Grand
 32
 Players select five numbers from 1 to 32.

Illinois
 Lucky Day Lotto
 45
 Players select five numbers from 1 to 45.

Indiana
 Cash 5
 41
 Players select five numbers from 1 to 41.

Louisiana
 Easy 5
 37
 Players select five numbers from 1 to 37.

Maine
 Gimme 5
 39
 Players select five numbers from 1 to 39.

Maryland
 Bonus Match 5
 39
 Players select five numbers from 1 to 39.

Massachusetts
 Mass Cash
 35
 Players select five numbers from 1 to 35.

Michigan
 Fantasy 5
 39
 Players select five numbers from 1 to 39.

Minnesota
 Gopher 5
 47
 Players select five numbers from 1 to 47.

Minnesota
 Northstar Cash
 31
 Players select five numbers from 1 to 31.

Missouri
 Show Me Cash
 39
 Players select five numbers from 1 to 39.

Montana
 Montana Cash
 45
 Players select five numbers from 1 to 45.

Nebraska
 Pick 5
 38
 Players select five numbers from 1 to 38.

New Hampshire
 Gimme 5
 39
 Players select five numbers from 1 to 39.

New Jersey
 Cash 5
 43
 Players select five numbers from 1 to 43.

New Mexico
 Roadrunner Cash
 37
 Players select five numbers from 1 to 37.

New York
 Take 5
 39
 Players select five numbers from 1 to 39.

North Carolina
 Cash 5
 39
 Players select five numbers from 1 to 39.

Ohio
 Rolling Cash 5
 39
 Players select five numbers from 1 to 39.

Oklahoma
 Cash 5
 36
 Players select five numbers from 1 to 36.

Pennysylvania
 Cash 5
 43
 Players select five numbers from 1 to 43.

Rhode Island
 Wild Money
 35
 Players select five numbers from 1 to 35.

South Carolina
 Palmetto Cash 5
 38
 Players select five numbers from 1 to 38.

South Dakota
 Dakota Cash
 35
 Players select five numbers from 1 to 35.

Tennessee
 Tennessee Cash
 35
 Players select five numbers from 1 to 35.

Texas
 Cash Five
 37
 Players select five numbers from 1 to 37.

Vermont
 Gimme 5
 39
 Players select five numbers from 1 to 39.

Virginia
 Moneyball
 35
 Players select five numbers from 1 to 35.

Virginia
 Cash 5
 34
 Players select five numbers from 1 to 34.

Washington
 Hit 5
 39
 Players select five numbers from 1 to 39.

Wyoming
 Cowboy Draw
 45
 Players select five numbers from 1 to 45

More lottery guides with explanations of how to calculate odds, payouts, and expected returns.
New York Lotto vs. Sweet Million
Keno Lottery Odds and Strategies
Lottery Odds Formula
Odds of Winning Hot Lotto
Random Lottery Number Generators
Cash 5 & Fantasy 5 Payouts
Since each state operates its pick5 lottery game independently, the prizes vary by jurisdiction. Some offer fixed amounts for the partial matches; some award prizes on a parimutuel basis, which means their values fluctuate from drawing to drawing depending on the pool of money available and the number of winning lottery tickets.
Game Variations
Some jurisdictions offer slight variations on the basic Pick 5 game.
 Some states offer a multiplier option for an extra $1 or $0.50. For each drawing, a multiplier is randomly selected from between 2 and 5, or is fixed at a certain value such as 2 or 3. If the numbers on your lottery ticket partially match the lottery drawing and you have purchased the multiplier option, then your prize for a partial match is multiplied by the multiplier value. The jackpots are never multiplied.
 In Virginia's Cash 5 game, players can wager $0.25, $0.50, or $1 on a set of numbers, just as long as the ticket totals $1. This means you can bet on four sets of numbers for $0.25 each, or two sets of numbers for $0.50 each. If you win a $0.25 wager, you are awarded one fourth the amount of the prize. Likewise if you win a $0.50 bet, you win half of the prize value. Probabilitywise, if you make four wagers your overall odds of winning are four times better, and if you make two wagers your odds are twice as good.
 In some states, if the top prize is not won, the jackpot amount rolls down to the 4/5 and 3/5 partial match prize levels. This is in contrast to most lotteries where the jackpot rolls over and is added to the next drawing's jackpot.
Comments
I play Gimme 5 in Vermont. When I was looking at the prize chart the other day, I noticed an irregularity in the prizes. The top prize is $100000 with odds 1:575757. The second prize is $250 with odds 1:3387. The way I calculate, since 575757/3387 = 170, then the first prize to second prize ratio should also be 170. That means the second prize should be $588 instead of $250, because with 100000/588 you get 170. I hope I'm explaining this clearly. The third prize is also wonky and it should be $18 instead of 7. The fourth prize is fine. What is the explanation for this?