If you are just picking random lotto ticket numbers then he is wrong - buying two tickets does not double your chances of winning. Here’s the math behind it:
Let’s call the chance that ticket #1 wins p1, and the chance that ticket #2 wins p2.
Let’s also note that those events are statistically independent (whether or not ticket #1 wins has no effect on whether ticket #2 wins) and they are not mutually exclusive (the range of possibilities includes exactly one ticket winning, both tickets winning, or neither ticket winning).
Given that, the probability that AT LEAST one ticket wins is not p1+p2, but rather p1+p2-(p1*p2). This is a basic formula from probability and statistics, the probability of the union of two events is equal to the sum of their individual probabilities minus the probability of their intersection (that is, the probability of both events occurring together).
If you just add p1 and p2, you are counting twice that tiny chance that both tickets happen to be winners. You can’t do that. Imagine the lottery was a single digit from 0-9, if you bought 11 tickets at random would you have a 110% chance of winning? Of course not.
So let’s suppose p1=p2=(10e-8, or 1/100,000,000).
The probability of winning if you buy two tickets = (10e-8)+(10e-8)-(10e-810e-8), which is 210e-8 - 10e-16.
Not exactly twice the odds of a single ticket, although in this case because p1 and p2 are so small, it’s really close.
Now, if you intentionally pick the numbers on your lotto tickets so that no ticket is a duplicate of any other ticket then buying two tickets does in fact exactly double your chances of winning, since you are specifically eliminating the chance of both tickets winning at the same time; the intersection has no chance of taking place and that little subtractive term drops out.