# UltraLotto 6/58 Probabilities | Odds and Prizes

TR Smith is a product designer and former teacher who uses math in her work every day.

UltraLotto is the largest jackpot game currently offered in the Philippines. It was introduced in February 2015 and is a 6/58 game, meaning players choose 6 distinct numbers from between 1 and 58. If they match all 6 numbers they win the jackpot, and if they match a subset they can win a smaller prize.

Like other lottery games in the Philippines (Lotto, MegaLotto, SuperLotto, and GrandLotto) players have the option of buying more than the basic 6 numbers per ticket. These are called system plays. If you buy N numbers, the price increases by a factor of (N Choose 6), the combination function, aka the number of groups of 6 you can make out of N objects. In UltraLotto you can buy up to 12 numbers per ticket. This is explained in following table. Structurally, UltraLotto is very similar to Brazil's Mega-Sena Lottery.

As the newest lottery game, UltraLotto has become the focus of much excitement and many players want to know how to calculate their odds of winning. Frequent lottery players often wonder if the system plays are worth the extra cost. All of these questions can be answered by computing the probabilities.

## Ultra Lotto System Plays

System Play
How Many Numbers You Select
How Many Different Groups of 6 There Are in Your Selection
Lottery Ticket Price Increase Factor
System 7
7
7
7
System 8
8
28
28
System 9
9
84
84
System 10
10
210
210
System 11
11
462
462
System 12
12
924
924

How to Use This Table: The base lottery ticket costs 20 Philippine pesos (Php) and lets you choose 6 distinct numbers from 1 to 58. If you buy a System 9 ticket, you can select 9 numbers and the price increases by a factor of 84, making it cost 1,680 Php. This is because within a set of 9 objects, there are 84 different subsets of size 6.

## How to Compute UltraLotto Probabilities

The Philippines UltraLotto awards prizes for matching 3/6, 4/6, 5/6, and 6/6 numbers. The probability of matching X out of 6 depends on how many numbers N you have selected, where N ranges from 6 to 12. The probability equation is

[(52 choose N-X) * (6 choose X)] / (58 choose N)

where the combination function (A choose B) is given by the formula A! ÷ [B! * (A-B)!]. For example,to calculate the probability of matching 6/6 with the base ticket, you let X = 6 and N = 6. This gives you

[(52 choose 0) * (6 choose 6)] / (58 choose 6)
= 1 / 40475358
= odds of 1 in 40,475,358

In other words, your chances of winning the UltraLotto jackpot are about 1 in 40.5 million with the basic ticket. As another example, your probability of matching 4/6 with a System 11 play is

[(52 choose 7) * (6 choose 4)] / (58 choose 11)
= 5405 / 613263
≈ odds of 1 in 113.46

The probabilities of matching X/6 for each system play are summarized in the table below.

System Play
Odds of Matching 3/6
Odds of Matching 4/6
Odds of Matching 5/6
Odds of Matching 6/6
Basic
1 in 92
1 in 2,035
1 in 129,729
1 in 40,475,358
System 7
1 in 56
1 in 907
1 in 37,792
1 in 5,782,194
System 8
1 in 37
1 in 472
1 in 14,455
1 in 1,445,549
System 9
1 in 26
1 in 273
1 in 6,556
1 in 481,850
System 10
1 in 20
1 in 171
1 in 3,346
1 in 192,740
System 11
1 in 15
1 in 113
1 in 1,864
1 in 87,609
System 12
1 in 12
1 in 79
1 in 1,111
1 in 43,805
System Play
Overall Probability of Winning Any Prize
Overall Odds of Winning Any Prize
Basic
0.011419368
1 in 87.5705
System 7
0.019137027
1 in 52.2547
System 8
0.029306177
1 in 34.1225
System 9
0.042051512
1 in 23.7804
System 10
0.057435094
1 in 17.4110
System 11
0.075462730
1 in 13.2516
System 12
0.096090095
1 in 10.4069

## Is It Worth the Money to Buy the System Plays?

A System 7 ticket in UltraLotto costs 7 times as much as a basic ticket with 6 number selections, but your overall odds of winning only improve by a factor of roughly 1.68

A System 8 tickect costs 28 times as much as a basic ticket, but the overall odds only improve by a factor of roughly 2.57.

The same is true for Systems 9, 10, 11, and 12. Playing more numbers increases the price by a huge factor, but the overall odds do not increase nearly as much. Only the jackpot odds increase proportionally to the price increase. The best way to increase your probability of winning is to buy several basic 6-number tickets with non-overlapping number selections. For example if you buy seven basic tickets with the numbers

{1, 2, 3, 4, 5, 6}
{7, 8, 9, 10, 11, 12}
{13, 14, 15, 16, 17, 18}
{19, 20, 21, 22, 23, 24}
{25, 26, 27, 28, 29, 30}
{31, 32, 33, 34, 35, 36}
{37, 38, 39, 40, 41, 42}

you will spend 7*20 = 140 Philippine pesos, the same as a System 7 play. But your probability of winning any prize will be about 7 times better, roughly 1 in 12.5. Those are much better odds than the System 7 play in which you choose seven different numbers with overall odds of 1 in 52. System plays are designed to milk more money out of players.

## UltraLotto Prizes

In UltraLotto the jackpot prize for matching 6/6 starts at a minimum of 50 million Php. The second prize for matching 5/6 is up to 280,000 Php. The third prize for matching 4/6 is up to 3,800 Php. The fourth prize for matching 3/6 is 20 Php, which is the cost of a standard ticket. All prizes are pari-mutuel, meaning that instead of being set at fixed amounts each drawing, the prize pool allocated for each prize level is split equally among the winners. The more winners there are, the less you get.

## UltraLotto Compared to Other Philippines Lotto Games

The Philippines lottery commission (Philippine Charity Sweepstakes Office) runs four other lotto games that are basically the same as UltraLotto except for the range of numbers and the bounds on prize amounts. All of them allow for the system plays discussed above.

• Lotto: A 6/42 game, i.e., players choose 6 distinct numbers from 1 to 42. The odds of winning the jackpot with a basic ticket are 1 in 5,245,786.
• MegaLotto: A 6/45 game, i.e., players choose 6 distinct numbers from 1 to 45. The odds of winning the jackpot with a basic ticket are 1 in 8,145,060.
• SuperLotto: A 6/49 game, i.e., players choose 6 distinct numbers from 1 to 49. The odds of winning the jackpot with a basic ticket are 1 in 13,983,816. Around the world, 6/49 games are one of the most popular lottery formats. See also Canadian 6/49 and New Jersey Pick 6.
• GrandLotto: A 6/55 game, i.e., players choose 6 distinct numbers from 1 to 55. The odds of winning the jackpot with a basic ticket are 1 in 28,989,675.

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