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Many people who play traditional Pick-5, Pick-6, or, in general, Pick-K lottery games

where each independent drawing consists of the random selection without replacement

and without regard to order of K numbers, are unaware of the fact that certain groups of

combinations offer statistical advantages compared to other groups of the same size. For

example, if you had wanted to purchase 10 tickets for the original Virginia Pick-5 game

(see my note at the end of this ad), then it would have been to your advantage, as far as

the probabilities of a 4- and a 3-number win are concerned, to play the 10 combinations

that appear below under the heading of Group 1 instead of those that comprise Group 2;

the likelihood of a 5-number win is the same for each group of 10 Pick-5 combinations.

Thus, you would have gotten more “bang for your buck”, so to speak, if you had spent

your money on the combinations in Group 1 rather than those in Group 2.

Group 1 Group 2

_______ _______

01 10 15 18 25 01 02 04 13 16

02 06 23 26 28 01 13 18 23 28

02 07 16 24 25 02 03 05 07 26

04 07 19 30 31 05 06 08 13 34

05 11 14 23 25 06 07 19 22 23

05 11 21 22 25 06 08 13 24 34

06 19 26 29 34 06 12 14 24 28

09 18 19 22 24 08 10 14 25 33

13 16 22 25 33 10 11 15 20 32

19 20 26 29 32 18 23 27 28 33

It’s not readily apparent that Group 1 provides the greater advantage. Actually, there are

many groups containing exactly 10 combinations that are statistically more favorable than

Group 1. Using software that I wrote (see my other Ebay ad entitled “Computer Software For

Traditional Pick-5, Pick-6, & PowerBall Lottery Games”), I have compiled groups of 5, 10, 15,

20, 25, and 30 combinations for the original Virginia Cash 5 game (maximum lottery number

34, no 2-number win) that may be regarded in each case, based upon probability criteria, as

the best of 200,000 computer generations. A PDF of these combination groups, including all

possible win scenarios, probabilities, and payoffs associated with each group, as well as

additional information about the original Cash 5 game, consists of 69 pages. I also have a

PDF pertaining to the Virginia Cash 4 Life PowerBall game that lists groups of 5, 10, 15, 20,

25, and 30 combinations that may be considered in each case as the best of 3000 computer

generations; the Cash 4 Life PDF is 24 pages long. Although these files contain information

that is specific to the original Virginia Cash 5 and Cash 4 Life lottery games, the mathematical

approach which is described can be applied to any similarly designed Pick-K and PowerBall

game. The PDFs will be shipped on a thumb drive (the thumb drive includes both PDFs.

Note:

The original Virginia Cash 5 game, which debuted in February of 1993, was a fixture of

the Virginia Lottery for many years. In October of 2020 the Cash 5 was altered in several

ways. One significant change was an increase in the maximum lottery number from 34 to

41, which reduced the one-play probabilities of a 5-number, a 4-number, and a 3-number

win by a divisor of approximately 2.69, 2.17, and 1.74, respectively. The newer version of

the Cash 5 also includes a 2-number win whose one-play probability is about 9.53 %; the

original Cash 5 did not offer a 2-number win. The payoffs for the new Cash 5 game, with

one exception, have also been modified. The most notable change concerns the jackpot

which, instead of being fixed at $100,000 for each winning ticket as was the case in the

earlier version, now starts at $100,000 and, if there is no winner, increases by a minimum

of $5000 for the next drawing. The jackpot continues to progress in this manner until there

is at least one winner. It’s important to mention that the jackpot is an estimate and that the

amount set aside as payment is divided equally among multiple winners, so that the actual

jackpot awarded to each winning ticket may be less than, equal to, or greater than $100,000.

A 4-number win is now worth $200 instead of $100, the cash prize for a 3-number win is still

$5, and the payoff for a 2-number win is $1.

The Virginia Lottery is correct in stating that the new version of the Cash 5 game offers the

player better odds of winning a cash prize compared to the original game. The overall one-

play probability of winning a new Cash 5 prize is approximately 10.39 %, whereas the overall

one-play probability of winning a Cash 5 prize based upon the original version of the game is

about 1.51 %. Keep in mind, however, that the original Cash 5 did not include a 2-number win.

If it had done so, then the overall one-play probability of winning a cash prize would have been

approximately 14.64 %.

One thing to remember is that whenever a state agency alters a lottery game, the primary

reason for doing so is to increase revenue, consequently the revision is almost always less

favorable to the player than the original version. Don’t confuse the favorability of a game with

its fairness. For practical, if not ethical, reasons, all state-sponsored lottery games are fair (in

other words, the drawings are independent and there is no bias associated with the selection

of the winning lottery numbers). These games are generally unfavorable to the player,

however. By choosing sufficiently large maximum lottery numbers, and by manipulating the

payoffs and the ticket prices, the state lotteries ensure that the expected value of a game to

the player, which is the expected one-play payoff minus the cost of a ticket, is negative. For

example, the expected value of the original Cash 5 game to the player was about - $ 0.52,

indicating that, on average, the Virginia Lottery made a profit of approximately $ 0.52 for each

ticket sold. If a 2-number win with a payoff of $1 had been added to the original Cash 5, then

the expected value of the game to the player would have been about - $ 0.38. Because of the

variable nature of the new Cash 5 jackpot, it is impossible to ascertain exactly the overall

expected value of this game to the player. Nevertheless, an approximation for this value can

be obtained in the following way. Select a random sample of the actual winning jackpots (for

a particular drawing, the actual jackpot is the amount paid to each winning ticket). If J denotes

the arithmetic mean, or average, of these sample jackpots, then an estimate of the expected

value of the new Cash 5 game to the player is $ (J - 610498) / 749398 (this mathematical

expression is specific to the new Cash 5). For example, if J = $75,000, $125,000, or $225,000,

then the expected value of the new Cash 5 game to the player would be about - $ 0.71,

- $ 0.65, and - $ 0.51, respectively. An algebraically equivalent approach is to take each

sample jackpot, call it J again (in this case J is an individual sample, not the average of the

samples), evaluate $ (J - 610498) / 749398, and then calculate the arithmetic mean of these

values.

There is another part to the new Virginia Cash 5 game called EZ Match that, for an extra

dollar per play, offers you a chance to win instant cash prizes ranging from $2 to $500. If any

of your Cash 5 numbers match any of the 5 randomly generated EZ Match numbers, you win

the cash prize shown next to that number. The original Cash 5 game did not have a feature

comparable to EZ Match.