I bought a civic about a week ago. When I went to test drive the car, it had to be boosted but ran fine once it was started. So I went ahead and bought a new battery for it. Well, I've been driving it around and mostly everything has worked fine until today. My girlfriend was driving and she says that all of a sudden, the radio shut off, the lights started dimming and eventually the car just shut off (all in the span of about 30 seconds). I went to boost and it fired up no problem again, but within 30 seconds of disconnecting the cables, the car died again. I suspect alternator issues BUT yesterday I was driving and all of a sudden, the speedometer stopped working and the check engine light came on. I never got around to checking the code but I did pull the fuse for the speed sensor, which was burnt out. Never got around to replacing the fuse either (it's been a hectic couple of days). So my question is, is it possible that these two issues are connected? could there be some underlying issue here? Also, am I right in thinking that the alternator is most likely fried?
Thanks in advance!

 

Aaaaand I just realized something haha. So the speed sensor and alternator are on the same fuse (duh). which explains the alternator issue (again, duh). But I still wonder, why did the fuse burn out? do these things just happen or is there something that would cause this to happen