If you flip a fair coin times, what is the probability that the only consecutive tail flips are flips and ?
A misunderstanding of a recent daily problem had me trying to figure our a general formula for flips, but as I started to look for that pattern, a clear approach escaped me.
The sequence of flips would have to end in regardless of the sequence length.
So as far as I can tell, I need to figure out the number of ways to fill the remaining 's with 's and 's such that no 's are adjacent.
I ended up doing this by hand up to 7 flips, finding 8 sequences.
My attempt was to eliminate from all possible adjoining strings those with adjacent 's.
Start with 's in the adjoining sequence ( ever single sequence passes the filter ).
there are adjoining sequences
Trying to remove the grouped 's I imagine the following:
We can only fill the boxes with so we only have 4 possible strings with them grouped.
Thus there are , sequences that pass the filter.
However, my methods seems to get confusing pretty quickly as the length of the sequence grows and we add 's to the sequence.
With 's , I have to not only worry about groupings, but also the sub groupings of 's. Overlaps seem to be everywhere in my started method of counting...and its all rather messy.
While I might be able to keep going for a bit , this continuation seems certainly seems like a failure for generalizing. That most likely is a result of my own limitations, but I'm still interested!
Any help leading me to water on this will be appreciated!