Find the probability that there are only or exactly 4 consecutive heads in 7 tosses. In my previous edit, I asked whether the case, e.g. $HHHHTHH$ (4 consecutive heads and 2 consecutive heads) is possible regarding the question, "Find the probability that there are only 4 consecutive heads in 7 tosses."
Now, I would like to ask more about whether the case as stated above can be applied to the following one, "Find the probability that there are exactly consecutive heads in 7 tosses."
In my opinion, this case should not be possible since the word 'only' or even 'exactly' has restricted the case to have just 4 consecutive heads. Am I right?
 A: Both formulations are not entirely clear. It’s possible to arrive at a most plausible interpretation, but only by eliminating other possible interpretations as less plausible.
The problem is that it’s not clear which other possibilities “only” or “exactly” is meant to exclude. This could be further tosses in the same run, further runs with as many consecutive tosses, or even, as you seem to have interpreted it, further consecutive tosses, even in shorter runs.
A clue to the most plausible interpretation lies in looking closely at how the phrase “$4$ consecutive tosses” is conventionally used. If we take it literally, we’d have to construe “consecutive” as a property of heads. Possible candidates for this property might be that a heads is “consecutive” if it is preceded or followed by heads, or that it is “consecutive” if it is preceded by heads. In the former case, HHTHH would be an instance of $4$ consecutive heads, and in the latter case HHHHHwould be an instance, whereas HHHH wouldn’t. This is clearly not how the phrase “$4$ consecutive heads” is conventionally used.
So if “consecutive” isn’t a property of individual heads, perhaps we can interpret it to refer to pairs of consecutive heads. But that doesn’t work either, since “$4$ pairs of consecutive heads” would include HHTHHTHHTHH, which is clearly also not how “$4$ consecutive heads” is conventionally used.
Rather “4 consecutive heads” is conventionally used to mean “a run of $4$ consecutive heads”. This is also how you seem to be interpreting it, as you have a run of $4$ consecutive heads in your example HHHHTHH. (Incidentally, that example also has exactly $4$ pairs of consecutive heads.)
But if “$4$ consecutive heads” is short for “a run of $4$ consecutive heads”, then “only/exactly $4$ consecutive heads” should be short for “a run of only/exactly $4$ consecutive heads”. That would speak in favour of interpreting “only/exactly” as limiting the number of consecutive heads in a run. Then “there are only/exactly $4$ consecutive heads“ could be taken to mean “there is a run of only/exactly $4$ consecutive heads”. If so, HHHHTHH would qualify.
All this is, of course, merely a speculative attempt at clarifying what is unclear. As has been stated in the comments, if possible it would be best to ask the author to clarify.
