How many trials are needed to ensure that the probability of winning
the lottery at least once is greater than one-half ?
1 - ( 1 - p )n > 1 / 2 → 1 - ( 1 / 2 ) > ( 1 - p )n
i.e. 1 / 2 > ( 1 - p )n → - log 2 > n * log ( 1 - p ).
Recall that c = nbr of ways to choose 6 numbers from 49 = 13,983,816
p = 1 / c = 1 / 13,983,816
and so 1 - p is just a tiny bit less than one.
So, log ( 1 - p ) < 0. [ Obvious from looking at a graph of y = log x ]
Dividing both sides of - log 2 > n * log ( 1 - p ) by a negative quantity reverses the sense of the inequality; and we get
n > ( - log 2 ) / log ( 1 - p ) ≈ 9,692,888
Thus we'd need more than nine million trials.