How lotto probability works

For a game where k different numbers are drawn from 1…N, the number of possible combinations is

C(N, k) = N! / (k! (N − k)!)

For 6/58 that is 40,475,358. One line therefore has a 1 in 40,475,358 chance of matching all six.

The chance of matching exactly r numbers with one line is

P(r) = C(k, r) · C(N − k, k − r) / C(N, k)

Buying m different lines for one draw gives m / C(N, k). Playing the same line for d draws gives 1 − (1 − 1/C(N,k))^d — still tiny.

A single number appears in any one draw with probability k / N, so over D draws you'd expect it about D · k / N times. Real counts wobble around that value by chance; that wobble is what "hot" and "cold" lists show.