How do I randomly pick something from a table with raritys?

One way of going about this is by duplicating the values in the table, like so:

local values = {"value1","value1","value2","value3"}

This way, however, can be long to type, so you could do something like this:

-- For each entry, the first value is the actual value, and the second is the 'weight' or chance.
local values = {
    {"value1", 2},
    {"value2", 1},
    {"value3", 1}
}

local SelectFrom = {}
local SelectedValue
for i, v in ipairs(values) do
    for i2 = 1, v[2] do
        table.insert(SelectFrom, v[1])
    end
    SelectedValue = math.random(1, #SelectFrom)
end

This works like the first method, but compresses the values, then decompresses them when selecting.

There may be other (better) ways; this is just the first one I came up with. It is also untested.

The first method is a pretty good one, and I'd go with that because it's the first one I thought of too. The alternative is to have a weight table, wherein the selection is done in part of a single random number out of 1.

local items = {
    {item1, 2},
    {item2, 3},
    {item3, 1},
    {item4, 1}
};
local max = 0;
for i=1,#items do max = max+items[i][2] end;
local selection,cnt = math.random(),0;
for i=1,#items do
    cnt = cnt+items[i][2]/max;
    if cnt > selection then selection = items[i][1] break end;
end;
if type(selection) == 'number' then selection = items[#items][1] end --Just in case

selection would at that point become your item. This is good because it can be mixed with other loot tables and have inheritance implemented with ease.