How lottery probabilities work

Counting combinations

When k distinct numbers are drawn uniformly from n numbers and order does not matter, there are C(n, k) = n! / (k! × (n − k)!) possible sets. The chance of matching one complete set is 1 / C(n, k). This assumes a fair draw and one valid standard entry.

P = 1 / C(n, k)

An example you can check

In a hypothetical draw of 2 numbers from 5, there are 10 possible pairs. One specified pair has probability 1 in 10. There are 6 pairs containing exactly one of its numbers, so exactly one match has probability 6 in 10. These are examples, not historical results.

Separate bonus pools

When the rules require an independent bonus selection for the jackpot, multiply the numbers of possible selections from each pool. A supplementary ball drawn from the remaining main balls is different: it may determine a lower prize category without adding a jackpot selection.

Jackpot, any prize and profit

The jackpot probability concerns one category. The chance of any prize adds the probabilities of mutually exclusive winning categories. Neither is the probability of making a profit. A return calculation also needs ticket cost, prize amounts, sharing and the applicable rules; a larger jackpot probability alone does not establish a better return.

What historical statistics can tell you

Frequencies and gaps describe a dated sample. Under independent draws, a number does not become due because it has been absent. Compare archives only within compatible rule periods and inspect sample size and source dates. An archive refresh check does not guarantee live ingestion.

Lottery is a category, Lotto is also a game

Lottery covers different number-draw formats. Italian Lotto is a particular game with wheels and its own prize categories; it is not interchangeable with SuperEnalotto or Oz Lotto. Choose the actual game before interpreting a probability.

Exact jackpot probabilities: five formats

Game