Probability of an event happening more than usual The whole question is: if an event normally happens 30% of the time, what's the probability of this event happening more than 30% of the time?
 A: This is a special case of the Binomial Distribution called the Bernoulli Distribution (where we have one trial and a fixed probability of success - $p$).
Let's use the random variable $X$ to denote the event. When $X=1$ this will represent the event happening, and when $X=0$, this will represent the event not happening.
The CDF of this distribution is $F_x(x) = 1-p$ if $x \in (0,1)$. This tells us that the probability of the event happening less than $30$% of the time is $1-0.3 = 0.7$ since $p=0.3$ in your particular example. And so the probability that this event happens less than $30$% of the time is $0.7$.
Therefore, the probability that this happens more than 30% of the time is simple $30$%.
As we can see from the PDF diagram below, the fact that the event is skewed in favour of the event not happening, this helps to explain the calculations above:

Additional Comments
I should make it clear that, that in the real world, things are rarely this simple. Some things to consider are:

*

*Repeated events do not always have a fixed probability of success every time.

*The probability of it happening more than a given percentage of the time may depend on the number of times that the event is repeated

*If we no longer events and instead look at other variables (for example, the probability that something will take more than a certain amount of time over a given percentage of time), this is now a continuous random variable, and will need to be treated differently as it follows a different probability distribution altogether.

Edit
As mentioned in the other answer, this does get a little more complicated if we repeat this experiment $n$ times.
This now becomes the binomial distribution, which has a different CDF. I do not find this particularly insightful as the solution isn't as "nice" (we end up with quite a messy function), but for completeness I will include this in my answer.
We observe the following PDF diagram showing the number of times the event occurs out of $100$ trials (for $n=100$):

This should, again, help with the visual intuition.
Now, we have a more complicated CDF with the formula $F_x(x) = I_q(n-x,x+1)$ which tells us that the probability that an event occurs less than $30$% of the time will be $Fx(0.3n)=I_q(0.7n,0.3n+1)$ where $I_q(a,b)$ is the "regularised incomplete beta function". This solves the sum mentioned in the other answer, however, like I mentioned - this overcomplicates the problem and it is much easier to think about what this probability will be after $1$ trial rather than $n$ trials.
A: Assume that you have $n$ trials.
Then, the number of successes, will be some element $k$ in $\{0,1,\cdots,n\}$.  Let $f(k)$ denote the probability that you have exactly $k$ successes, under the assumption that the probability of success in each trial is $(0.3)$, and under the assumption that the trials are independent of each other.
Then, $$f(k) = \binom{n}{k} 0.3^k ~0.7^{n-k}.$$
Let $$A = \lfloor (0.3) \times n\rfloor.$$
That is, $A$ equals the largest integer $~\leq (0.3 \times n).$
Then, the probability of having more than $30\%$ of the $n$ trials succeed is
$$\sum_{k=A+1}^n f(k) = \sum_{k=A+1}^n \left[\binom{n}{k} 0.3^k ~0.7^{n-k}\right].$$
