Can you explain this statement and mathematically verify if it's correct? "“A nuclear attack is inevitable. It’s the ultimate problem of mankind. If there’s a ten percent probability that something will happen in a year, there’s a 99.5 percent probability that it will happen in fifty years. But if you can get that probability down to three percent, that reduces the probability to only seventy-eight percent in fifty years. And if you can get it down to one percent, there is only a forty percent probability in fifty years. That’s a truly worthwhile goal—it could literally make all the difference in the world.”
 A: The way to calculate the probability of an event ever happening is to instead think of the problem as the probability of no event happening ever.
If there is a 10% probability for some event, then the probability of it not happening is 90%.
The probability of a 10%-per-year event not happening in 2 consecutive years is .9 * .9 = 81% (so, a 19% chance of the event happening at some point). The probability of a 10%-per-year event happening after 50 years then is the cumulative probability of 50 years of no-event: .9^50 = .005. Therefore a 99.5% chance of happening.
1-.90^50 = 99.5%
1-.97^50 = 78%
1-.99^50 = 40%
So these numbers from the statement are correct.
However, estimating the actual annual percent risk is quite tricky. The statement 

A nuclear attack is inevitable

depends heavily on what the actual annual risks are, the time frame that one considers, and how that risk changes with passing years of non-nuclear weapons usage (i.e., if the probability is not independent over time). It's also quite unlikely that the true risk is anything close to constant: during the peak Cold War years the use of nuclear weapons was seriously considered and on a couple occasions came very close to use. Basic statistical probabilities aren't much use to us here.
