Sounds a lot like the Toro I got a couple years ago.
Take off the necessary screws to get the cover off, and look for where the fuel line comes down from the fuel tank. That goes into the carb. Tip the snow blower so the carb is fairly high up. Take the bolt off the bottom of the carb (probably 7/16 or 1/2). That bolt should have a very small hole in it, where the gas goes though. Hole is aobut 1/3 of the way from the hex head end.
Pull the bowl off the carb, and clean it out. See if there's any particles of junk. Water.
Work the float up and down a couple times. Near the pivot end (axle) is the needle that allows gas into the carb. Poke that with a needle or small screw driver if it doesn't move with the float.
tip the machine down, and see if a little gas pours out of the carb. Put the carb bowl back on, snug down that nut that holds it on.
While you have the machine apart it's a really good idea to put in an inline fuel filter. I happened to have one around, your mower shop nearby (or hardware store) might have one you can put in. I had to cut about 3 inches of fuel line out to make space for the filter, and then use hoseclamps. I had a whole bunch of specks of black stuff in the fueltank, and I had to take the carb apart several times before I put the fuel filter in.
And that is what I found with mine...... it's still miserable, but it will start after awhile.
50:1 sure seems lean to me. I mix mine 32:1 and nevermind what it says on the gascap.