As the title says,
How do I calculate the probability of the sum of random variables being more than a number, knowing that the number is higher than a constant. With every random variable having an exponential distribution with the same λ.
For example:
I throw apples on a bag until the total weight of the bag exceeds the 5 kilograms. The weight of each apple is a random variable which has an exponential distribution, with λ = 1/3. Calculate the probability that the total weight is more than 7.
Have you ever heard of a Poisson process? That is essentially what you're asking about. If people walk into a shop at rate $\lambda$, the time between people is $\text{Exp}(\lambda)$, so you can imagine drawing exponential random variables and letting those be the distances between marks you put on a ruler from $t=0$ to $\infty$ to signify visits. You're interested in the probability that there are no marks between $t=5$ and $t=7$, so what you're asking is the probability that (returning to Poisson process language) no one enters the shop between $t=5$ and $t=7$. The number of people who enter during this time is a $\text{Poisson}(2\lambda)$ random variable, so your answer is $e^{-2\lambda}$.