How lucky you are if you win the lottery?

Many people find in the lottery the chance to change their life, but the reality is that the odds are so low that most people can be playing for hundreds of years without ever get closer to the prize. A thought usually goes through the mind of the players is: "If there are people who won the price, why cannot I win it too?", which I do not doubt, but let us look at the numbers and see how lucky you would be if you won the lottery.

Suppose you are playing a typical lottery that consists of 49 numbers, where 6 numbers are randomly selected in each drawn. Other lottery games with different draws can be calculated using the same ideas presented here.

In order to calculate the probability of correctly predicting the 6 numbers, it is necessary to introduce some basic concepts. First, it is important to know that this probability is related to the number of possible combinations that could be generated, in a chance of 1 in the number of combinations.

In probability theory, there are two concepts that should not be confused: Combinations and Permutations. If we take into consideration the order, then we are talking about permutations, otherwise we speak of combinations. In the lottery usually does not matter the order in which the numbers appear, so we will focus, for now, in the combinations, and its calculation formulas.

There are two types of combinations, those where repetition is allowed and those where it is not, as is the case under study, where the numbers are chosen one by one, without repetition, and everyone has an equal chance of being chosen.

There is a formula to calculate combinations without repetition, however, I propose to do a little reasoning that leads us to the formula. Let us analyze the lottery example consisting of 49 numbers to select 6.

Basically, to calculate the total number of combinations where the numbers are not repeated, it is necessary to compute the total amount of chances to choose six numbers orderly, and then divide the result by the total number of permutations for the 6 numbers previously selected.

Let us first obtain total orderly possibilities to select 6 random numbers. There are 49 different but equally likely ways of choosing the first number. To select the second number, there are now only 48 different ways, which is the amount of available numbers. Thus for each of the 49 ways of choosing the first number there are 48 different ways of choosing the seconds, so the total number of ways of choosing the first two numbers can be calculated as the product of 49 and 48, which is equal to 2352.

Analogously, to select the third number, there is 47 different ways, and 49x48x47 = 110544 possibilities of choosing the first three numbers. This continues until the sixth number has been selected, giving the final calculation, 49x48x47x46x45x44 = 10,068,347,520.

As explained above, from the fact that the order of the 6 numbers is not significant, we must divide the number obtained in the previous step, by the number of possible ways to permute the six selected numbers. The reasoning for its calculation is similar as above.

The first number can be placed in any of the 6 positions. Once the first number have been placed, the second can take any of the 5 remaining positions, and so on. At the end of the process, it gives 6x5x4x3x2x1 = 720 possible permutations. The mathematical operation where the first k integer numbers are multiplied, is called factorial and it is written k!, in this particular case 6! = 720.

Dividing 10,068,347,520 by 720 gives 13,983,816, so the probability of predicting all the 6 numbers correctly is 1 in 13,983,816. If the lottery is drawn daily, theoretically, it takes more than 38,000 years to drawn all possible combinations. As can be seen, the chance of winning the jackpot is extremely small, so you should not feel frustrated by misfortune.

Note that 49x48x47x46x45x44 can be written using the factorial notation as:

49! / 43! = 49! / (49 - 6)!

and the final formula as:

49! / 6!x(49 - 6)!

Generalizing this formula, we can say that the number of different combinations to select r numbers out of n, is:

which is denoted with large brackets and is called "binomial coefficient". Using this formula, it is easy to calculate the number of possible combinations for any lottery, and thus the odds of predicting all numbers.

If the draw has two different set of numbers to be chosen, such as EuroMillions, where you must predict a combination of 5 numbers (from 1 to 50) and 2 stars (from 1 to 11 ), just calculate the total combinations of both sets independently, and then multiply the results. For this particular case it gives:

(50x49x48x47x46 / 5x4x3x2x1) x (11x10 / 2x1)

= (254,251,200 / 120) x (110 / 2)

= 2,118,760 x 55

= 116,531,800

So, the odds to win the EuroMillions is 1 in 116,531,800

Although there are many computer programs that claim to predict the next lottery numbers, the reality is that the randomness of the machines used for the draws has been well tested, giving no opportunity for the design of algorithms, that based on historical data, could predict the next draw numbers.

My advice is that despite everything you have read here, you still want to play the lottery and keep alive the hope that one day you can be the winner, do it as a hobby, not as an alternative to improve your finances, in which case I advise you to work harder, it is the safest way to achieve it.