The Odds Of Winning From EZ2 Lotto

EZ2 lotto is a system 2/31 which means 2 balls are drawn from numbers 1 to 31. To play the game, a player picks 2 numbers from 1 to 31. To win the game, the 2 numbers a player picked must correspond to the numbers drawn either in any order or exact order. A player must indicate in the bet card if he is playing in any order or exact order. Playing in any order is like playing 2 sets of combinations.

The Odds Of Winning EZ2 Lotto

If you play your numbers in any order, your chance of winning is 1:496 or 0.202% probability (calculated as 1 ÷ 496 = 0.202%). 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. If you increase the numbers that you play, you also increase your chance of winning by the probability rate just mentioned. For example, if you have 10 tickets equivalent to 10 combinations, you increase your chance 10 times, i.e. 10:496 or 1:50.

Let's Do The Math

For the sake of credibility, I will talk a bit about how to arrive at the number of possible combinations. You can skip this section if you are not interested in the formula.

First, let's differentiate two things. In Statistics, there is a term called combination, and another is called permutation. The difference between the two is that, in combination formula, the order of the numbers does not matter. In permutation, the order of the numbers is significant. In the case of EZ2, however, we are going to use only the combination formula and a simple multiplication method, instead of the permutation formula.

Combinations (C)

Combinations refer to all possible combinations of numbers regardless of their order. Therefore, the combination AB is the same as BA. For example, 12-21 and 21-12 is just but one possible combination.

All in all, in a system 2/31, there are 465 possible combinations. For the reason that 2 tumblers are used to draw the winning numbers, it is possible to form double numbers, e.g. 22-22. There are 31 possible double-number combinations – 1,1 up to 31,31. Thus, 465 + 31 = 496 possible combinations.

The formula to arrive at the number of combinations is a bit complicated. (See the figure on the right).

To read the formula is something like this: n choose k equals the n factorial all over k factorial times n-k factorial where:

n is the maximum count of balls, 31 in the case of EZ2
k is the number of balls in a combination, 2 balls in the case of EZ2
! means factorial, e.g. 5! is equivalent to 5 x 4 x 3 x 2 x 1.

Let's substitute.

₃₁C₂ = 31! ÷ ( 2! (31-2)! ) = 31! ÷ ( 2! x 29! ) = 465

This is the same as ₃₁C₂ = 31! ÷ 29! ÷ 2! = 465

A shortcut is ₃₁C₂ = (31 ÷ 2) x (30 ÷ 1) = 465

In Excel, the formula is COMBIN(31,2). In OpenOfficeCalc, the formula is COMBIN(31;2). The only difference is the use of comma and semi-colon. The factorial formula in both Excel and OpenOfficeCalc is FACT(number).

You may be wondering how the complicated formula can be simplified as in the shortcut above. Well, it has something to do with cancelling what's common in the nominator and the denominator.

Ordered Combinations (OC)

Permutations refer to all possible combination of numbers in every possible order. In this sense, the combination AB is not the same as BA. Therefore, 12-21 and 21-12 are two separate permutations. A permutation can either allow or not repetition of numbers. If repetition is allowed, it means that a number can be combined to all of the other numbers including itself.

For EZ2 lotto, the order of numbers is significant. The repetition of numbers is allowed. To calculate how many permutations (i.e ordered combinations) can be formed from 1 to 31, the answer is 961. The formula is easier than the combination and permutation formulas as far as precalculus is concerned.

OC = n^r

where OC is the number of all possible permutations; n is the maximum number of balls in a tumbler, in this case is 31; and r is the count of numbers a combination has, which in this case, 2.

To substitute the formula with values,

OC = 31²

or

OC = 31 x 31

OC = 961

To learn more about combinations and permutations, this online tutorial site explains the formulas quite easily.

Increasing The Odds

With the aid of probability data, some combinations can be eliminated to increase the odds. What if you want to find out which number – 1 to 31 – wins frequently.

Below is the data collected from EZ2 Lotto results drawn from January 2011 to September 2013. The lotto numbers are ranked according to the number of times the number won (frequency of winning). If you are going to look at the rate of probability, it looks there's not much difference. You can interpret this as all numbers having equal or fair share of winning. On the other hand, between 224 and 167, the frequency of winning of the numbers 10 and 18 respectively, the difference of 57 recurrences can look significant. The recurrence is another way to look at the probability data. The figures refer to the number of draws. For example, number 10 recurs (likely wins on an average) every 13 draws.

Total		5880
Rank	Lotto#	Freq	%	Recur
1	10	224	3.8%	13
2	04	212	3.6%	14
3	16	205	3.5%	14
4	30	201	3.4%	15
5	01	200	3.4%	15
5	25	200	3.4%	15
7	15	199	3.4%	15
8	08	198	3.4%	15
8	14	198	3.4%	15
10	06	194	3.3%	15
10	31	194	3.3%	15
12	23	193	3.3%	15
13	03	192	3.3%	15
13	22	192	3.3%	15
15	13	191	3.2%	15
16	26	190	3.2%	15
17	05	189	3.2%	16
18	02	188	3.2%	16
18	21	188	3.2%	16
20	29	187	3.2%	16
21	28	186	3.2%	16
22	27	183	3.1%	16
23	17	180	3.1%	16
24	12	179	3.0%	16
25	11	178	3.0%	17
26	24	177	3.0%	17
27	20	175	3.0%	17
28	19	174	3.0%	17
29	07	173	2.9%	17
29	09	173	2.9%	17
31	18	167	2.8%	18