Probability question related with exponential distributed random variable In an infection under treatment with antibiotics, a certain bacterium’s lifetime in hours is described by an exponentially distributed random variable with parameter 0.15. a) What is the probability the bacterium won't survive for 3 hours? b) If the bacterium survives 12 hours, what is the probability the bacterium last 1 hour longer? * * * For part a I know that the density of exponential distribution is $λ\cdot\exp(-λ\cdot x)$. So I should integrate $λ\cdot\exp(-\lambda\cdot x)$ with the lower bound of 0 and upper bound of 3. The calculation ended up being 0.36237.

Sure you _can_ integrate, and you did so correctly, but did you also know what is the _cumulative distribution function_ for an exponential distribution?

For $X\sim\mathcal{Exp}(\lambda)$ :$$\mathsf P(0
So for $\lambda=0.15$, $~\mathsf P(0
So that's the answer to (a), which you got the hard way.

The answer to (b) is now a simple application of the definition of conditional probability. (Although, again there is an easier method. Are you aware of the "memoryless" property and what it means?)