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# Texas Lottery

## Introduction

The Texas Lottery opened for business on May 29, 1992 (). The percentage of ticket sales paid back as prizes is 63.1%. Of the rest, 27% goes towards education, 9.5% towards expenses, and 0.4% in unclaimed wins goes mostly towards Veterans (). Not every game pays back 63.1%. In fact, the return ranges from a low of 44% to a high of 78%, so choose your game carefully.

The Texas Lottery has an unusual rule that players of games where the grand prize is in the form of an annuity must choose whether to accept the annuity or a lump sum payment at the time of purchase. Why they bother everybody with this question, I have no idea.

• MEGA MILLIONS USA -
• POWERBALL USA
• NEW YORK LOTTO USA
• FLORIDA LOTTO USA

The rest of this page contains my analysis of the several ways to play the Texas Lottery. However, for those who just want the executive summary, here is the overall return of each game:

• Pick 3: 48% to 50%
• Daily 4: 44.3% to 50%
• Cash 5: 50%
• Lotto Texas: 50% to 51.8%
• Texas Two Step: 50%
• Triple Chance: 54.40%
• All or Nothing: 56.00%
• Powerball: 50%
• Mega Millions: 50%
• Scratch Cards: 60% to 77.8%

Next, here is my quick advice on how to play each game where there is some decision to be made other than picking numbers:

• Pick 3: Play straight "exact order" bets, as opposed to box "any order" bets, or on a total of 0, 3, 24, or 27.
• Daily 4: Play straight "exact order" bets, as opposed to box "any order" bets, or on totals of 3 or less or 33 or more.
• Lotto Texas: Invoke the Extra option, which carries a higher return than the base game.
• Scratch Cards: The return is greater as you go up in price. Do not play games with a trademarked name.

Finally, here is my general advice for playing any lottery game:

1. Don't play in the first place. Every state lottery offers terrible odds. With few exceptions, it is the worst bet you can make.
2. For any game involving choosing numbers, I suggest the Quick Pick option. Players that pick their own numbers are at greater risk to get short changed on a jackpot by having to share it with many other players who picked the same numbers for the same reasons you did.

What follows is my analysis of each game.

## Pick 3

The is played four times a day (except Sunday). The player and Lottery each pick three numbers from 0 to 9, with replacement. Bets can be \$0.50, \$1, or \$5. There is also a "Sum it Up" bet, based on the total of the Lottery's three numbers.

The player can choose to play "exact order" or "any order." It is harder to win if the numbers must match in order, but you stand to win more. The probability of winning, and the win, playing "any order" depend on the whether the player chooses three different numbers (such as 1-2-3) or a pair of the same number (for example 1-1-2).

The following table shows the win, probability of winning, and return for all three ways to play. The right column shows the expected return is 50% playing "exact order" and 48% playing "any order."

### Pick 3 Return Table

Bet Pays Combinations Probability Return
Exact order 500 1 0.001 0.500
Any order (1-2-3) 80 6 0.006 0.480
Any order (1-1-2) 160 3 0.003 0.480

The next table summarizes the Sum it Up bet. The average total is 13.5, so the further you bet from that, the more you will get paid if you win. The right column shows the maximum return is 50% on totals of 0, 3, 24, and 27.

### Pick 3 — Sum it Up

Bets Pays Combinations Probability Return
0 or 27 500 1 0.001 0.500
1 or 26 166 3 0.003 0.498
2 or 25 83 6 0.006 0.498
3 or 24 50 10 0.010 0.500
4 or 23 33 15 0.015 0.495
5 or 22 23 21 0.021 0.483
6 or 21 17 28 0.028 0.476
7 or 20 13 36 0.036 0.468
8 or 19 11 45 0.045 0.495
9 or 18 9 55 0.055 0.495
10 or 17 7 63 0.063 0.441
11 or 16 7 69 0.069 0.483
12 or 15 6 73 0.073 0.438
13 or 14 6 75 0.075 0.450

## Daily 4

The is played every day except Sunday. The player and lottery each pick four numbers from 0 to 9, with replacement. Bets can be \$0.50, \$1, or \$5. There are also bets for matching two specific numbers and the sum of all four numbers.

The following table shows the probability of winning each prize, the average win, and return for each way to play. There is a separate line for each way to play a box bet, according to how the numbers may repeat. The numbers in parenthesis are just an example. For example, 1-1-2-2 would be any two pairs of numbers.

### Daily 4 Return Table

Bet Pays Combinations Probability Return
Straight 5000 1 0.0001 0.50
Box (1-1-1-2) 1200 4 0.0004 0.48
Box (1-1-2-2) 800 6 0.0006 0.48
Box (1-1-2-3) 400 12 0.0012 0.48
Box (1-2-3-4) 200 24 0.0024 0.48

As you can see from the return column, the best bet is a straight bet, where the number must match in order, at a return of 50%.

There are three Pair bets (Front Pair, Middle Pair, Back Pair), which are bets on matching just two specific numbers. They pay \$50, have a 1% chance of winning, and offer a return of 50%.

The next table summarizes the Sum it Up bet for the Daily 4. The average total is 18, so the further you bet from that, the more you will get paid if you win.

### Daily 4 — Sum it Up

Bets Pays Combinations Probability Return
0 or 36 5000 1 0.000 0.500
1 or 35 1250 4 0.000 0.500
2 or 34 500 10 0.001 0.500
3 or 33 250 20 0.002 0.500
4 or 32 142 35 0.004 0.497
5 or 31 89 56 0.006 0.498
6 or 30 59 84 0.008 0.496
7 or 29 41 120 0.012 0.492
8 or 28 30 165 0.017 0.495
9 or 27 22 220 0.022 0.484
10 or 26 17 282 0.028 0.479
11 or 25 14 348 0.035 0.487
12 or 24 12 415 0.042 0.498
13 or 23 10 480 0.048 0.480
14 or 22 9 540 0.054 0.486
15 or 21 8 592 0.059 0.474
16 or 20 7 633 0.063 0.443
17 or 19 7 660 0.066 0.462
18 7 670 0.067 0.469

The right column shows the maximum return is 50% on totals of 0, 1, 2, 3, 33, 34, 35, and 36.

## Cash 5

Cash 5 is a simple \$1 daily game where the player and lottery each pick 5 numbers from a range of 1 to 37. Wins are for matching 2 to 5.

The percentage of ticket sales returned to winning players is a minimum of 50% (source: ). The top three awards are awarded a sub-share of the 50% share returned to players. Wins for matching 2 are always \$2. The following table shows the sub-share awarded to each win.

• Match 5: 40.15%
• Match 4: 18.08%
• Match 3: 41.77%
• Match 2: \$2

The following table shows the average prizes under these rules, assuming there is at least one win for catching five.

### Cash 5 Return Table

Catch Average
Win
Combinations Probability Return
5 \$47,677.52 1 0.000002 0.109378
4 \$134.19 160 0.000367 0.049254
3 \$10.00 4,960 0.011379 0.113791
2 \$2.00 49,600 0.113788 0.227577
1 \$- 179,800 0.412483 0.000000
0 \$- 201,376 0.461981 0.000000
Total   435,897 1.000000 0.500000

It often happens that nobody matches all five numbers. When that occurs, the 40.15% sub-share for catching 5 is rolled down into the share for catching 4. The following table shows the average prize for each win, assuming that nobody matched all five.

### Cash 5 Return Table

Catch Average
Win
Combinations Probability Return
5 \$0.00 1 0.000002 0.000000
4 \$432.17 160 0.000367 0.158632
3 \$10.00 4,960 0.011379 0.113791
2 \$2.00 49,600 0.113788 0.227577
1 \$0.00 179,800 0.412483 0.000000
0 \$0.00 201,376 0.461981 0.000000
Total   435,897 1.000000 0.500000

The probability of any win is 12.55%.

## Lotto Texas

Lotto Texas is a \$1 game where the player and Lottery each pick six numbers from 1 to 54. There is also an "extra" bet the player can make that bumps up the smaller wins. Draws are on Wednesdays and Saturdays. If the player wins the jackpot, it will be paid as a 30-year fixed annuity, unless the lump sum option is taken.

According to the , the portion of ticket sales directed towards prizes are directed as follows:

• Catch 6: 40.47%
• Catch 5: 2.23%
• Catch 4: 3.28%
• Catch 3: Fixed prize of \$3

The following table shows the probability of winning, average prize, and contribution to the return for all possible outcomes, assuming there was at least one jackpot winner.

### Lotto Texas

Catch Average
Win
Combinations Probability Return
6 \$10,452,253.68 1 0.00000004 0.404700
5 \$1,999.81 288 0.00001115 0.022300
4 \$50.07 16,920 0.00065512 0.032800
3 \$3.00 345,920 0.01339365 0.040181
2 \$0.00 2,918,700 0.11300892 0.000000
1 \$0.00 10,273,824 0.39779140 0.000000
0 \$0.00 12,271,512 0.47513972 0.000000
Total   25,827,165 1.00000000 0.499981

The probability of any win is 1.41%.

In the event nobody wins the jackpot, the 40.47% of ticket sales will trickle down to pay winners who catch 5. The average win for catching five when there are no jackpot winners will be \$38,292.36.

The next table shows the pay table, probability of winning, and contribution to the return for the Extra option. The player may not bet the Extra option by itself but it is an optional bet if the player is already playing the base game.

### Lotto Texas — Extra Bet

Catch Average
Win
Combinations Probability Return
6 \$0 1 0.00000004 0.000000
5 \$10,000.00 288 0.00001115 0.111510
4 \$100.00 16,920 0.00065512 0.065512
3 \$10.00 345,920 0.01339365 0.133936
2 \$2.00 2,918,700 0.11300892 0.226018
1 \$0 10,273,824 0.39779140 0.000000
0 \$0 12,271,512 0.47513972 0.000000
Total   25,827,165 1.00000000 0.536977

The probability of any win is 12.71%.

The bottom line is the return is 50.00% on the base game and 53.70% on the Extra bet. That said, my advice is that if you must play, your overall return will be higher if you make the Extra bet.

## Texas Two Step

The is a \$1 game with a draw every Monday and Thursday. The object is to match the draw of four White Balls from a range of 1 to 35 and one Bonus Ball from a separate pool of 35 balls. The player can win by matching the Bonus Ball or at least three of the White Balls.

According to the , the portion of ticket sales directed towards prizes are directed as follows:

• Catch 4 with Bonus Ball: 23.78%
• Catch 4 without Bonus Ball: 2.79%
• Catch 3 with Bonus Ball: 0.34%
• Catch 3 without Bonus Ball: 4.6%
• Catch 2 with Bonus Ball: 3.04%
• Catch 1 with Bonus Ball: Fixed prize of \$7
• Catch 0 with Bonus Ball: Fixed prize of \$5

In the event nobody wins the jackpot, then the money will roll over to the next drawing. The jackpot reset amount is \$200,000.

The following table shows the average win for each prize, the probability, and contribution to the total return.

### Texas Two Step

Catch Bonus
Ball
Average
Win
Combinations Probability Return
4 Yes \$435,792.28 1 0.0000005 0.237800
4 No \$1,503.81 34 0.0000186 0.027900
3 Yes \$50.25 124 0.0000677 0.003400
3 No \$20.00 4,216 0.0023006 0.046000
2 Yes \$19.97 2,790 0.0015224 0.030400
2 No \$- 94,860 0.0517625 0.000000
1 Yes \$7.00 17,980 0.0098112 0.068678
1 No \$- 611,320 0.3335807 0.000000
0 Yes \$5.00 31,465 0.0171696 0.085848
0 No \$- 1,069,810 0.5837662 0.000000
Total     1,832,600 1.0000000 0.500026

The probability of any win is 3.09%.

## Texas Triple Chance

Texas Triple Chance is a game based on a draw of 10 numbers out of 55. While the Lottery will draw ten numbers, the player picks seven only. For a \$2 bet the player gets three plays. There is one draw per day, at 10:12 PM, except Sundays.

The table below shows how much the player wins if he matches 3 or more numbers. The return column is the product of the win, probability, and 3/2. The reason for multiplying by 3/2 is the player gets three tickets for each \$2 bet. The bottom right cell shows a rate of return of 54.42%.

### Triple Chance

Catch Win Combinations Probability Return
7 \$100,000.00 120 0.000001 0.088702
6 \$500.00 9,450 0.000047 0.034926
5 \$25.00 249,480 0.001229 0.046103
4 \$5.00 2,979,900 0.014685 0.110134
3 \$2.00 17,879,400 0.088107 0.264322
2 \$- 54,979,155 0.270930 0.000000
1 \$- 81,450,600 0.401377 0.000000
0 \$- 45,379,620 0.223625 0.000000
Total   202,927,725 1.000000 0.544186

The probability of any win is 10.41% per single game. The probability of at least one win over three games is 28.08%, assuming independently drawn numbers.

## All or Nothing

is a misnomer of game. The player does indeed win the most by hitting all or none of his numbers, but there are also consolation prizes for getting close to either extreme. The way it works is the player will pick 12 numbers from a range of 1 to 24. The Lottery will randomly do the same. A ticket costs \$2. The Lottery will have a draw four times a day, except Sundays.

The table below shows how much the player wins for matching 0-4 or 9-12. The return column is the product of the win, probability, and 1/2. The reason for multiplying by 1/2 is to show the relative return based on a \$2 bet. The bottom right cell shows a rate of return of 55.98%.

### All or Nothing

Catch Win Combinations Probability Return
12 \$250,000.00 1 0.000000 0.046225
11 \$500.00 144 0.000053 0.013313
10 \$50.00 4,356 0.001611 0.040271
9 \$10.00 48,400 0.017898 0.089492
8 \$2.00 245,025 0.090611 0.090611
7 \$- 627,264 0.231963 0.000000
6 \$- 853,776 0.315727 0.000000
5 \$- 627,264 0.231963 0.000000
4 \$2.00 245,025 0.090611 0.090611
3 \$10.00 48,400 0.017898 0.089492
2 \$50.00 4,356 0.001611 0.040271
1 \$500.00 144 0.000053 0.013313
0 \$250,000.00 1 0.000000 0.046225
Total \$- 2,704,156 1.000000 0.559823

The probability of a win per game is 22.03%.

## Multi-State Games

Texas participates in part of both multi-state lotteries, the Powerball and Mega Millions.

According to Lottery regulations, approximately 50% of money bet is returned to winning players (Source: and ).

Please see my Powerball page for more information on that game. A Mega Millions page is coming soon.

## Scratchers

Scratch card games are simple games on a card. The player scratches to reveal his prize. At the time of this writing, scratch cards were available at costs of \$1, \$2, \$3, \$5, \$10, \$20, and \$50.

The Lottery doesn't implicitly state the percentage of money returned to the player. However, they kindly indicate the number of tickets printed and the number of each win, so it is easy to deduce the number of losers. With that, it is easy to calculate the return percentage. Let's look at the \$1 game, as an example. They tell us everything we need to know, except the numbers of losing tickets. They tell us there are 11,104,650 tickets in the game. The sum of winning tickets is 2,372,575, which must mean there are 8,732,075 losers.

We can now set up a return table for the game, as follows:

### Lucky 9's

Win Tickets Probability Return
\$1,000 113 0.000010 0.010176
\$500 251 0.000023 0.011302
\$100 1,549 0.000139 0.013949
\$50 1,866 0.000168 0.008402
\$20 49,461 0.004454 0.089082
\$10 135,589 0.012210 0.122101
\$5 135,602 0.012211 0.061056
\$3 258,972 0.023321 0.069963
\$2 592,353 0.053343 0.106686
\$1 1,196,819 0.107776 0.107776
Loser 8,732,075 0.786344 0.000000
Total 11,104,650 1.000000 0.600493

The lower right cell shows a return of 60.05%. The probability of any win is 100% - 78.63% = 21.37%.

However, not all games have the same return. As a rule of thumb, for any given denomination, returns are in a tight range. As you pay more for a ticket, the return percentage goes up. Here is the average return of some games I sampled, by denomination, not counting trademarked games:

### Texas Scratch Card Average Returns*

Bet Average
Return
\$1 60.12%
\$2 65.01%
\$3 65.93%
\$5 68.01%
\$10 71.00%
\$20 74.97%
\$50 77.81%

*: Does not include games with a Trademarked name.

You may be wondering at this point, what do I have against including Trademarked games? A lot. I find they depress the return by up to 8%! In all fairness, many of the Trademarked games offer non-cash prizes in a second chance drawing you must enter online to win. As an example, the has a return of 64.56%, which is 3.45% less than the average for a \$5 ticket of active non-trademarked games at the time of this writing. However, the player can win a trip to the World Series or other "Merchandise Prize Packs" in the second-chance drawing. These second chance drawings seem to only happen for trademarked games and not all of them.

Next, I would like to mention that Texas is nice enough to have a vague policy of retiring games early if all the big winners are claimed. Here is how the Texas Lottery words it:

"Game closing procedures may be initiated for documented business reasons. These games may have prizes unclaimed, including top prizes. Game closing procedures will be initiated when all top prizes have been claimed. During closing, games may be sold even after all top prizes have been claimed."

If we assume that the instant all of the biggest prizes are sold the Lottery will immediately stop selling tickets (a big "if"), then that increases the return by the ratio of the jackpot amount and the total number of tickets. For example, in the Lucky 9's game above, the return would go up by \$1,000/11,104,650 = 0.009%.

However, this is a big "if" because:

• The rules say the Lottery "may" initiate closing procedures, which implies it also may not.
• Tickets may be sold anyway during closing procedures. I have no idea how long that phase lasts.
• Jackpot winners may delay cashing their tickets, which may give the illusion that there are winning tickets left unsold, when there are not.

However, if we assume a best case scenario that the Lottery always does closes a game immediately when there are no jackpot winners left, then I find it increases the return by up to 4.09%, with an average of 0.62%. The benefit tends to be more for the more expensive tickets, which have fewer jackpots. In general the value of this rule is more for games where the ratio of jackpot tickets is small.

Finally, I'm sure some people will not appreciate my mentioning this, but there is a potential advantage play in scratch card games. The Texas Lottery is nice enough to indicate how many tickets for each win have already been cashed. If there is a game that is almost sold out, as evidenced by the small wins, with a high ratio of large wins still unclaimed, then it may mean the remaining unsold tickets are rich in big winners. The same principle as card counting in blackjack. I'll leave the details to the reader (don't you hate it when I say that?).

• — Official web site.
• — The fine print of the game rules.
• — Page on the Texas Lottery

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