The rule 3. Know your chances
The probability of a jackpot in any lottery is, usually, one formula. And here is the calculation of the probability, eg, closing at least one line in the lotto is very non-trivial and would take a whole article, maybe, and not one. Therefore, in fact, the chance of getting some money in the lottery is higher due to the fact, that most lotteries have additional prizes in addition to the main one. But I will focus on the jackpot for ease of assessment..
Let's admit, we bought a lottery ticket with a random set of numbers
The same number of balls are drawn during the drawing., and if the numbers on them coincided with the numbers on the ticket (in any order, it is important!), then we won. The probability of such a win is calculated as follows:
The number of combinations without taking into account the order is called in mathematics the number of combinations, and if you know and understand the formula for calculating it, then from this article you, likely, don't learn anything new. If you are not a mathematician, then it will be easier to use the online service, for example with this. Similar services (and formula, underlying their work) offer to set two numbers:
- n - the total number of possible options for one item. In our case, the object is a ball, and there are so many balls, how many numbers are in the lottery, more on this below.
- k - number of items in one sample. In our case, how many balls the lottery draws and how many numbers are in the ticket (supposed, that these quantities are equal).
so, if we have a lottery with a draw 5 balls, but in total in the lottery 50 balls with numbers from 1 to 50, then the probability of winning in it will be equal to one to the number of combinations for k = 5 and n = 50, i.e:
Consider a more complex case - the popular American Powerball lottery, in which the jackpot value exceeded one billion dollars. According to the rules, there is a basic sample of 5 numbers (from 1 to 69), and also one additional number (from 1 to 26). Need to get everyone match 6 numbers, to win.
It's not hard to understand, that the chance of getting the first set is equal to one to the number of combinations for k = 5 and n = 69 (i.e 11 238 513), and the chance to "catch" the last ball is 1 to 26. To get everything at once, these chances need to be multiplied, because events must happen at the same time:
In other words, if 300 million people will buy tickets, then only one will win. It shows, why the jackpot is often not won at all: lottery organizers just don't print as many tickets, so that among them the winning.
