The exersice is: The waiting time in a line for a person who arrives to get vaccinated is exponentially distributed with the expected value of 20 minutes. John is waiting on the line for 15 minutes already to get the vaccine and he is not yet been called. What is the probability that his turn to get the vaccine will happen in the next 10 minutes?
Do I need to take the Memorylessness property in consideration here or is it enough to calculate F(25)-F(15)? That is, to use the cumulative distribution function of the exponential distribution for the values 25 and 15 and substract it? Is there a rule of thumb for this?
You can calculate the conditional probability $\dfrac{F(10+15)-F(15)}{1-F(15)}$ in general
but the memoryless property of the exponential distribution tells you this is $F(10)$ as a shortcut