All these answers are good and correct. I would add "to depend (upon/on) someone/something": Examples: "I counted on (depended on) the tow line (tested to forces of 1200 newtons) to safely maneuver my yacht every day." A VERY BROAD & GENERALIZED rule when learning English is that verbal phrases and prepositional phrases sound more natural, less formal and less "foreign/exotic/special" than a single-word verb. Examples: verbal phrase vs. single verb "to count on" = "to depend" "to dig up" = "to excavate" "to screw up" (very informal) = "to equivocate" "to come in" = "to enter" "to get up" = "to rise" There are thousands more examples of this, but what I am saying is that "count on" is perfectly useful but less formal.

If you count on somebody then you can rely upon them. If you count on something happening you can rely on the fact that it will happen.

Count on = expect