George I can help you out, but I don't think you will like the cost.
Your first step is to determine how many watt-hours you need in a 24 hour period. So let's say you want to burn that 50 watt light for 8 hours. 8 hours x 50 watts = 400 watt hours. Now to account for system inefficiency multiply that by 1.5. 1.5 x 400 = 600 wh This is how much power you will need to generate every day on average.
Next piece of information you need is your solar insolation measured in sun hours on your shortest days in winter. You have to use worst case or shortest days of winter. Don't know where you are but the is usually around 2 or 3 hours. For our example let's say you are in Boston which has an insolation of 3 hours in December.
Ok to find the solar panel wattage is straight forward, take your daily wh / Sun Hour so 600 wh / 3 hours = 200 watts
Next step is the size of the battery. First determine the system voltage, we select 12 volts in your case. Next we need to multiply your daily adjusted usage by 5. The reaon is we never want to discharge a battery more than 20% in any given day to maximise the battery life of at least 5 years, and have reserve capacity to carry us through a cloudy day or two. So 5 x 600 wh = 3000 wh reserve capacity. From here is straight forward WH / V = AH, 6000 wh / 12 volts = 500 AH @ 12 volts
Lastly you will want a MPPT charge controller between the panel and battery. The minimum size is straight forward Panel wattage / system voltage, 200 watts / 12 volts = 17 amps, round up to 20 amps
Now for the fun part, cost estimation
Panel cost = $4 per watt = $4 x 200 = $800
Battery cost = $250 per Kwh = $250 x 3 Kwh = $750
20 Amp MPPT Charge Controller = $350
Misc hardware and wiring = rougly $200
Total equipment cost = $2100
Labor = ??????
If your electricity cost 14 cents per Kwh you are spending $2100 + labor to replace 6 cents worth of electricity per day. Those batteries have to be replaced about every 5 years, or less if kept in less than ideal conditions.
