I've been asking myself this from a video-game that I play but I don't know how to solve it.
A "monster" has 50% chance of "respawning" every 5 minutes after it's been killed.
So after 5 minutes, 50% chance the respawn has occured, after 10 minutes, 75% chance, after 15 minutes, 87.5% chance etc…
What is the average time for the respawn to occure ?
I think it can be determined because we have a trend that reaches 1 at infinity.
Thank you.
Best Answer
Let get some more precise notation on the problem.
The monster has $p=\frac{1}{2}$ chance of spawning after 1 time step (we could get back the minutes by multiplying by 5 in the end). So it has $p\cdot p=\frac{1}{4}$ chance of spawning exactly after 2 time steps. Let's denote $X(t)$ the probability of it spawning exactly after $t$ steps. Then this probability can be written as the sum of $X(1)$ to $X(t)$. Ex after 3 times step: $$ X(1) + X(2) + X(3) = 87.5\%$$ As you said, one can see that the cumulative probability $X(1) + X(2) + \cdots + X(t)$ will tend to 1.
The average (expected value) is just then weighted sum of thoses probabilites. We can compute $$ \sum_{t=1}^\infty t \cdot X(t) = \sum_{t=1}^\infty \frac{t}{2^t} = 2 $$ (There are many methods for computing this sum let me know if you need some guidance on it)
So we get that the expected time step for the monster to spawn is 2. Which mean in average, we expect it to spawn every 10 minutes.