Flipping rigged coin, calculating most common number of flips between heads You have a rigged coin that if flipped, has a $10$% chance of landing heads up. Let's say that we flip this coin $1000$ times. We know that it should land heads-up approximately $100$ total times, but what about the number of flips it takes in-between head-outcomes? For instance, if we get heads on flip #2 and flip #6, that would be $4$ total flips in-between. What should be the most often occurring number of flips in-between for heads given this situation? Why?

The probability of taking $k$ flips to obtain heads is $(1-p)^{k-1}p$, where in this case $p=0.1$. Independent of $p$, this is maximal for $k=1$.