That is so weird. Where I live, Israel, they actually have different phone numbers (police - 100, medics - 101, fire dept. - 102), and they come only if you call them specifically (if at all). Of course if something really bad happens you call the police first and they call everyone else, but police arriving for someone who fell of their bike? Really? What a weird place.
Same in my location, a neighbor had a small fire in the wall due to plumbing soldering: police, firetrucks and ambulance. Then extra police kept stopping by (I guess they were bored, and the paramedic chick was cute and flirty).
I admit that this is probably the consequence of living in a less populated state, but when they ask the nature of your emergency here (NM), the dispatcher typically makes a decision of what response is needed.
All three are sent for most vehicular accidents for traffic control (state police, county sheriff, or city), fire suppression (volunteer FD, usually), and medical services (sometimes volunteer, especially in the county). Judging by the comments below, I suspect there's an advantage to living in an area that doesn't have tax money to blow. ;)