Lottery System 2/31: A Probability Study

Featuring PCSO's EZ2 Lotto (Philippines)

This report studied lottery results from PCSO's EZ2 Lotto, now called 2D Lotto. Studies were done for different periods such as: 2004 – 2008, 2008 – 2009, 2009 – 2013, and random results onwards. Other probability reports are also available such as:

6/58 6/55 6/49 6/45 6/42 6D 4D 3D 2D

ABOUT

This page is mainly about a lottery system 2/31 wherein two winning numbers are picked from 1 – 31. The subject of the study are drawing results from PCSO's EZ2 Lotto, now called 2D Lotto. Part I discusses briefly the general information about this lottery. If you are already familiar about this type of lottery, you may proceed to Part II; or jump to specific studies individually posted on this blog. Part II discusses the findings in studying the results of EZ2 Lotto, aka 2D Lotto.

SCOPE

The probability study covered results from 2004 and onwards. There are separate studies for 2004 – 2008 results and 2008 - 2009 results. The former is when the drawings of EZ2 were conducted once a day. The latter is when the drawings were conducted twice a day. There are also separate studies covering draw results from 2009 onwards; this is when the drawings were already done three times a day. For the later studies, it covered results up to the date as shown following this. The study is a work-in-progress; thus, some statistics may change as new results are added.

Part I

What Is A Lottery System 2/n?

A few basic information

A lottery system 2/n picks two winning numbers from a group of consecutive numbers. For example a 2-digit lottery, which is the most common, picks 2 numbers from 0 – 9. In the Philippines, a lottery system called EZ2 comprises numbers 1 – 31. If only one tumbler is used to draw the winning balls, repeating winning numbers are not possible. If two tumblers are used, wherein each tumbler contains, say 10 balls numbered 0 – 9, repeating numbers are possible. For example, both tumblers can draw the #08.

In other countries, like Canada, this game is commonly called Pick 2 or 2-Digit game wherein 2 numbers from 0 - 9 are drawn. This is called lottery system 2/10.

How To Play and Win Pick 2

To play Pick 2, you wage on 2 numbers from 0 - 9 in order to form a 2-digit number. For example, pick 1 and 2 to play the number 12. If your numbers matched the numbers that were drawn, you win a prize. In some games, if you matched either number, you win a minor prize. You can also wage on double numbers, e.g. 8-8.

To win Pick 2, like in Ontario, you win $2 if you matched the first digit. You win $99 if you matched both digits. For example, if you played 12, and the result of the draw was 13, you matched the first digit, which is 1; therefore you win $2. If the result of the draw was 12, you matched both digits, therefore, you win $99. If the result of the draw was 21, you don't win any prize because for Ontario's Pick 2 lottery, you always play for the exact order the winning numbers are drawn.

How To Play and Win EZ2 of the Philippines

To play EZ2 Lotto in the Philippines, you wage on 2 numbers from 1 - 31 to form a combination of 2 numbers. For example, pick 12 and 21 to play the combination 12-21. If your numbers matched the numbers that were drawn in exact order, you win the major prize, which is PHP4000. If your numbers matched the numbers drawn in no particular order, you win the minor prize of PHP2000. However, you have to specify your option in the game card or selection slip whether you are playing for no particular order (called jumbled play) of the winning numbers or for the exact order (called straight play). Jumbled play means you are playing for the two possible combinations. For example, if your numbers are 12 and 21, you are playing for both 12-21 win and 21-12 win. If you opted the straight method and played 12-21, you win the major prize if the results are 12 and 21. You do not win a prize if the results are 21 and 12. By default, you always play for the exact order.

You can wage on double numbers for EZ2, for example, 12–12. PCSO's EZ2 Lotto uses two tumblers to draw the two winning numbers — one winning number from each tumbler. Recently, PCSO rebranded EZ2 Lotto to 2D Lotto; though it does not refer to only 2 digits 0–9. For the sake of clarity, this report will refer to EZ2 Lotto as Lotto 2/31.

Trivia: In the Philippines, jumbled play is called rambol or rambolito, based on the English word ramble or rumble. Perhaps, the Filipino who invented the word, confused the word jumble with rumble or ramble.

How The Numbers Are Drawn?

There are 2 tumblers. Each tumbler contains balls numbered 1 - 31. One ball is drawn from each tumbler using air pressure or blower. For the Pick 2 system, each tumbler contains 10 balls numbered 0 - 9. The numbers are not digitally drawn for the reason that no automated system can exactly duplicate randomness.

How much do you wage for EZ2?

A ticket equivalent to one play costs only 10 pesos. If you play for two combinations, you pay 20 pesos. The price remains the same whether you are waging for a straight or jumbled (rambolito) play. You can, however, double or even triple your wage for a single play. For example, you can wage 20 pesos for a 12–21 number ticket. If you win straight, you win times 2 of the 4000-peso prize.

How many possible combinations can be formed for a lottery 2/31 such as EZ2 Lotto?

If you are playing for the straight method, i.e. where order matters, the number of possible combinations is 961. If order matters, it means that, say 12-21 is not the same as 21-12. If order of the numbers do not matter, the possible combinations are 496. This means, with our example, 12-21 is the same as 21-12. To verify the results, in Excel or Google Sheets, use the function =PERMUTATIONA() and =COMBINA() respectively.

Part II

EZ2 Lotto: A Probability Study

What's Your Chance of Winning?

In this section, we explore the numbers behind the numbers of a lottery system 2/31 by sampling actual drawing results from EZ2 Lotto (Now called 2D Lotto in the Philippines). The first study sampled data from July 2004 up to August 2008 lottery drawings. This is the period when EZ2 drawings were done once daily. It comprised 1,492 actual EZ2 lotto results.

The next study sampled results drawn from August 2008 up to August 2009. This is the period when EZ2 drawings were conducted twice daily. It comprised 732 actual EZ2 Lotto results. The two studies were then compared to find out any significance.

To determine if probability findings from the two past periods would remain consistent in future periods, the study sampled results from later drawings. All in all, about 6000 results were covered in multiples studies.

CHAP 1

The Odds of Winning

To win or not to win

First Things First

Odds of Winning The EZ2 Lotto

There are 961 possible two-number combinations that can be formed from 1 – 31 if order matters, which means AB is not the same as BA. If order does not matter, which means AB is the same as BA, there are only 496 possible combinations. If you play your numbers in any order, your chance of winning is 1:496 or 0.202% probability. If you play your numbers in exact order, your chance of winning is 1:961, or 0.104% probability. Playing your numbers in any order gives you better chances of winning.

Sampled Data Used In Analysing The Results

Statistical analyses of the data are based on draw results from the EZ2 lottery system. The data covers results from July 2004 up to a certain date as stated. Some analyses cover a period of one year, while others cover multiple years.

Initially, the studies covered draw results from July 5, 2004 to August 24, 2008. This is the period when drawings of EZ2 were done only once a day. It covers 1492 results.

The second set of the basis of studies covered the results from August 25, 2008 up to August 29, 2009. This is the period when drawings of EZ2 were done twice a day – one in the morning and one in the evening. It covers 732 results. Together, they make 2224 results.

The third set would be results from August 2009 onwards, which is when the drawings of EZ2 have been done three times a day – one in the morning, one in the afternoon, and one in the evening. At present, the draws are done at 2:00 PM, 5:00 PM and 9:00 PM.

The sampled data used are enough to give significant results because in Statistics, you only need 1200 sampled population to achieve a (plus-minus) ±3 margin of error.

Methods Used To Analyse The EZ2 Draw Results

The methodology in analysing the EZ2 draw results is simple arithmetic. A simple counting is used to calculate the recurrence or frequency of events, e.g. winning frequency of each number. To get the percentage, you simply divide a number from a larger number, e.g. the ratio of odd and even to winning numbers. Perhaps, the most complex formula used are figuring out the ideal sample size and the probability of winning the EZ2 lotto based on all possible combinations. These calculations, however, need not be done repeatedly for the reason that the results will remain unchanged.

In the worksheet or tool that you can download on Google Sheets, the formulas are more complicated. However, you need not worry about these formulas because they were already done for you.

Winning Frequency of Each Number

Methodology

The following study is based on more than 6000 actual draw results consisting of the winning numbers from the results of EZ2 drawings from 5 July 2004 onwards. The data were divided into three periods: 

1. July 2004 - August 2008 – the period when drawings were done only once a day; 
2. August 2008 - August 2009 – the period when drawings were done twice a day;
3. August 2009 onwards – drawings have been done thrice a day. 

To calculate for the frequency of winning, the number of times a number had won is simply counted. In spreadsheet (Google Sheets), the function COUNTIF(data reference, data to count) was used.

Winning Frequency of EZ2 Numbers 2004 - 2008

Following is the chart that shows how many times each EZ2 lotto number in 2004 - 2008 had won.

The Top, Middle and Bottom Numbers

Let's classify the winning numbers into three: the tops, the bottoms, and the middles. Based on the calculated data above, 

The top winning numbers were:

16 • 14 • 08 • 05 • 17 • 23 • 25 • 06 • 21 • 12 • 11

The middle winning numbers were:

07 • 19 • 28 • 26 • 10 • 09 • 20 • 24 • 04 • 01

The bottom winning numbers were:

13 • 03 • 27 • 29 • 18 • 15 • 30 • 22 • 31 • 02

Question: If we are to play the top winning numbers all the time, what's the guarantee of winning?

To find out, we have to compare the results of the 2004-2008 period with those of the succeeding period.

Top and Bottom Numbers Compared: Previous Period vs Next Period 

Following compares the winning performances of each EZ2 lotto numbers. The chart shows the ranking of the lotto numbers based on each frequency of winning. Ranking was used so that we can compare the top, middle and bottom winning numbers of the previous period with those of the succeeding period. Top winning numbers were ranked 21-31; while the bottom winning numbers were ranked 1-10; and the rest were the middle rankings.

Observations

Notice that most numbers that were previously top winning went down either to the bottom or middle rankings or vice versa. For example, lotto #16 was top winning previously, but in the next period, it was in the bottom. On the other hand, lotto #31 was in the bottom previously, but it soared the highest in the next period.

Summary

Following is the summary of the comparison's result.

• From top to bottom, there were five: 06 • 12 • 14 • 16 • 17
• From top to middle, there were three: 05 • 08 • 11
• From bottom to top, there were three: 31 • 22 • 15
• From bottom to middle, there were two: 27 • 02
• From middle to top, there were five: 04 • 07 • 19 • 26 • 28
• From middle to bottom, there were three: 09 • 10 • 24

The following EZ2 numbers maintained their standings:

• Remained on top, there were three: 21 • 23 • 25
• Remained neutral, there were two: 01 • 20
• Remained bottom, there were five: 30 • 29 • 18 • 13 • 07

Conclusion

• Only 3 top numbers remained on top. Therefore, not all top winning numbers will remain top.
• Half of the bottom numbers remained in the bottom; while three soared to top; and two advanced a bit higher.
• Five of the average performing numbers soared to the top.
• Therefore, whether the numbers were performing well or not or just average, they can be top winning numbers as well in the succeeding period. While others would remain in their current standing; the rest tends to slow down.

The Next Questions Are...

• How long can a top number remain on top?
• When can a bottom number or a middle number soar to the top?

To answer these questions, we have to monitor the standing of each number on a periodic basis.

Strategies To Try

• Play 5 numbers from the middle ranking numbers.
• Play 3 numbers from the bottom ranking numbers.
• Play 3 numbers from the top ranking numbers.

The challenge here, however, is which numbers do you pick. 

Rolling Top Winning Numbers

In this subject, we are going to look at the top winning numbers at a different perspective. What I did was to calculate the number of times a number has been winning for every 30-draw period. With these data, we might be able to figure out how long can a top winning number stay on top; or in other words, when will it cease to win once it has reached the top.

For every 30 draws, based on the 2004-2009 data, a number can win up to 9 times. There's no incident that a number had won more than nine times. All of the numbers 1-31 had experienced being at the bottom marks winning only 1-3 times per 30 draws.

What is a 30-Draw Period?

The 30-Draw Period is a rolling period. It means that, for every draw (or daily in case of the 2004-2008 data), the figures are calculated based on the last 30 draws, the current one included. So, if today is January 30, the last 30 draws (assuming draw is done once daily) cover those of January 1 - 30. The next day, January 31, the last 30 draws would be January 2 - 31; then January 3 to February 1; and so forth and so on. With this method, you have a real time idea which numbers are going top or bottom. Another term to describe it is: which one is currently trending.

Following is a screenshot of my worksheet. It's a tally of the winning frequency of each number for every 30-draw period. It also shows how many times each number became top winning, bottom, or neutral as presented under the columns 9 to 1. These standings are based on the 2004-2009 data.

Top, Middle and Bottom Numbers

First, let's define what makes a number top, middle, or bottom number.

• Top — Numbers marked 7, 8 and 9. These are numbers that won 7-9 times within a 30-draw period.
• Middle — Numbers marked 4, 5, and 6. These are numbers that won 4-6 times within a 30-draw period.
• Bottom — Numbers marked  1, 2, and 3. These are numbers that won 1-3 times within a 30-draw period. All numbers had won at least once; thus, we have no zero mark.

General Observations

Based on the data just presented, the following are my observations.

• A number can only win up to 7–9 times within a 30-draw period; maximum 9, often times 7. So, this means, if a number has already won 9 times from the last 30 draws, most likely, it is not going to win again in the next draw. If 8 times, the chance of winning again is very slim. Out of 7 numbers, only 2 might win again. Only #10 and #16 made it to the 9 mark; and it only happened once in 2224 draws. 
• Each number 01 - 31 had a fair share of being a bottom number (see column Bot).
• Twelve numbers never made it on top. These were: 5, 6, 8, 12, 13, 18, 20, 24, 26, 27, 29, 31. These numbers either won only once, twice or 3 times for every period calculated.
• Nineteen lotto numbers made it to the top. Among the top numbers, only 10 and 16 reached 9 winnings; while 3, 4, 21, 22, and 25 reached 8 winnings. The rest garnered 7 wins. These top numbers are those shown under the columns 9, 8, 7.

Profile of the Top Numbers

There were only two EZ2 lotto numbers that won 9 times from the period 2004-2009: 10 & 16.

• 16 — Number 16 made it to the top on 11/26/07 with 7 wins. It won again in the next draw for its 8th win. After 11 draws, it won again for its 9th win. It remained on top for 59 periods. However, the other side of the story is that, #16, after it hit 9 wins, it did not win again for the next 46 draws. When it won again on 01/26/08, #16 dropped to bottom 1. 
• 10 — Ten made it on top on 12/29/08 for its 7th win. On the same day, it won again to make its 8th win. It won again the next day making its 9th win. It remained on top for 21 periods; after which, its score dropped to 5, then to 3. Since it won the 9th time, it took 45 draws for #10 to win again.

Only 7 numbers reached the 8th time winner standing: 03, 04, 10, 16, 21, 22, 25.

• 03 — Number 03 made it early on top. Just after 69 draws since the launch of EZ2, it hit the 7 mark on 09/13/04. Immediately after, it won again on the next draw making it to the 8 mark. It stayed on top for 20 days; after which, its mark dropped down to 4, then to 2, then to the bottom 1. It took 91 draws for #3 to win again. 
• 21 — The number 21 reached 7 wins on 06/13/06. After 4 draws, it won again to make 8 winnings. It remained on top for the next 21 periods; after which, its winning dropped to 5 for the next 7 draws; then to 4 for the next 21 draws. It took 44 draws for #21 to win again. 
• Number 21 hit the 7 mark again on the 08/27/08; then on 08/30 to hit the 8 mark. It stayed on top for 32 periods; after which its score dropped to 3. It took 52 draws in order for #21 to win again.
• 25 — Lotto number 25 hit the 5 mark on 06/19/06. It won for 4 consecutive draws climbing its rank to 6, then 7. After resting for just a day, it won again to hit the 8 mark. It remained on top for 58 periods. After winning 8 times, it did not win again for the next 54 draws. When it won again, its rank dropped to one.
• 04 — Number 04 was very auspicious when it made to the top. It hit its 7th mark on 06/13/09 but it dropped to 6 after four days. It had  It had won 3x within 5 days hitting its 8th win on 6/19/09. It remained a top number for 43 periods; after which it dropped to 5 for only 2 periods to win again in the next draw. Since its 8th win, it took 22 draws for #04 to win again.
• 22 — Number 22 hit the top with its 8 wins two days before the lottery's transition to 3-times draw per day. It was a pompyang win—22 • 22.

Number Profiles Continued

Let's explore a few numbers; and see what the data are trying to say. This will also guide you how you can interpret that data on the worksheet just presented.

Exploring Lotto #25

One of the most favourite numbers in the calendar is 25 because it is associated with the most favourite holiday: Christmas. So, let's analyse the number 25.

In the comparative chart, #25 ranked top in both periods. It ranked #3 in 2004-2008; #4 in 2008-2009. Overall, it ranked #1 with 160 wins. Like any other number, most of the time, #25 is at the bottom for every 30-draw cycle (1332 on the worksheet). At other times, in the middle (738 times). Though rarely to be on top, compared to other numbers, it is one of the auspicious ones (82 times). Though it never reached the 9th mark, it reached the 8th mark twice, and the 7th mark five times. On the other hand, each time #25 wins, it stayed at the bottom marks (96 times). At other times, it reached the middle mark (55 times). What does this mean?

Each time #25 wins and its standing is at the bottom 1, expect that it is going to win again to rise to the bottom 2; then to 3, until it reaches the 6th mark. For the reason that it topped only 7 times, expect less that it will hit the 7th mark—time for #25 to rest and wait for its standing to anything lower than 6.

Let's drill down further. Based on 30-draw cycles, #25 ranked top 82 times with only 7 wins. This means, that #25 did not win 75 times out of the 82 draws (82 - 7). [see columns Top, 7, 8]. Therefore, each time #25 hit the top slots, it is going to rest for awhile.

On the other hand, #25 also ranked in the middle 738 times with 55 wins. This means, once #25 hits below the 6th mark, it may win again, provided it maintains its middle mark (4 to 6 wins).

Let's look at the bottom figures. Lotto #25 won 96 times while it was at the bottom. However, the sad story is that, out of 1332 draws, it did not win 1236 times.

To conclude, in case you can monitor #25 using the 30-draw cycles, observe the following:

• While #25 is at the bottom positions, there's a 7.20% chance that it can win again. [96 / 1332]
• While #25 is at the middle position, there's a 7.45% chance that it can win again. [55 / 738]
• When #25 reached the top position, there's an 8.53% chance that it can win again. [7 / 82]

Other Observations

Using the 30-draw cycles, each time EZ2 lotto #25 —

• drops to the bottom 1, there's a strong probability that it can win again within the next 9 draws, the next draw included. The chance is 89%.
• at the bottom 2, the chance to win again in the next 9 draws is 50%; if at the bottom 3, the chance is 39%.
• in the middle 4, the chance to win again in the next 9 draws is 54%; if at the bottom 5, the chance is 60%.
• in the middle 6, the chance to win again in the next 9 draws is 83%.
• in the top 7, the chance to win again in the next 9 draws is also 83%; if in top 8, the chance is 50:50. 

Findings

In terms of percentage, the winning probability of the numbers ranges from 2.9% to 3.9% (or 169 to 222 counts as of this writing). This means that each number had a fair share of winning. In other words, any number is as good as the other. If the data are to be graphed linearly in terms of percentage, it would look like almost a straight line. This means further that, in time, as the population or number of records grow larger, the line on the graph will flatten out. Further, therefore, to come up with significant trending results, a smaller set of records, say results from the last 2 weeks, would suffice.

Heat Map: 2004 – 2009

In this section, we explore the winning frequency of numbers as a pair. How often does a winning pair win in a span of 2224 draws? You will be surprised with the results.

The Heat Map Data

The following illustration presents the winning frequency of each of the 961 possible winning combinations. The grid presents the first ball in the first column; the 2nd ball in the topmost row. The blues are those that had won more than twice. The oranges are those that had won only twice or less. The whites are those that had not won (zero values).

The Famous Doubles (Pompyang)

The winning frequency of the doubles, or pompyang in Filipino, are presented in the grid with a border around it starting from 01-01 down diagonally to 31-31. Below summarises how each pair fared in 2004–2009. [Correction. The illustration below should have stated 2009 instead of 2008.]

Observations

• Most likely, for every 2000 results, a combination can win only up to 8 times maximum.
• A double (e.g. 12–12, 8-8, 29-29) can win only up to 6 times maximum. Most doubles win up to 3 times.
• Based on the database (not shown here), a double can win at no particular time. In other words, they are all unpredictable.
• Probability of a double combination to win:
• 2004 – 10 occurrences over 181 draws, 5.5%
• 2005 – 9 occurrences over 362 draws, 2.5%
• 2006 – 12 occurrences over 358 draws, 3.35%
• 2007 – 11 occurrences over 359 draws, 3.0%
• 2008 – 15 occurrences over 488 draws, 3.1%
• 2009 – 16 occurrences over 476 draws, 3.4%
• 2004–2009 overall – 73 occurrences over 2224 draws, 3.28%
• If a double combination can occur only up to 5%, therefore, non-double combinations can occur up to 95% of all draws.
• 97 combinations did not win at all (zero values in the heatmap)
• Only 5 combinations, out of 2224 draws, had won 8 times.

Proof of Randomness

Based on actual data as presented by the grid previously mentioned, 97 of the 961 possible combinations did not win over 2224 actual draws. If each combination has an equal chance of being selected in each draw, what is the probability that any particular combination goes unselected across all 2224 draws?

The calculation:

• Probability a specific combination wins in 1 draw: 1/961
• Probability it does not win in 1 draw: 960/961
• Probability it does not win in 2224 draws: (960/961)^2224 = 9.87%

What does this mean?

• Expected non-winners: 961*9.87% = 94.8
• Observed: 97
• Z-score: -0.04 below the mean, which is zero.
• Verdict: Consistent with randomness

With 2,224 draws, observing 97 non-winning combinations is almost exactly what random chance predicts. The Z-score of -0.04 means that the results of the draws are essentially right on target—well within the normal range of variation you'd expect from a truly random system. The Z-score must be less than 1.96 to be consistent with randomness.

What if there were only 1000 draws?

To prove that the lottery results were complete random, let's assume that there were only 1000 draws held instead of 2224.

• Probability it does not win in 1000 draws: (960/961)^1000 = 34.99%
• Expected non-winners: 961 * 34.99% = 336 combinations
• Z-score: -16.35
• Random assessment: Extremely non-random

You may want to read the complete explanation here and the conclusion here.

Apologies

I have written a comprehensive report about EZ2 on this page. Apparently, more than 80% of what I wrote here was completely gone. I don't know what happened. So, at this point, I am reconstructing the results of my studies.