The Red Cup

So, the first thing you should realize... The Red Cup is

Big!

Red Cup

The typical red cup is 18 oz. (previously 16 oz. but many stores are phasing out 16 oz. cups in favor of these larger versions.)

To give that some perspective:
Let's say you've had 3 Red Cups of beer, how many "drinks" have you had?
The Answer: Depends. If you have a 4.5 % alcohol beer then you've had 4.5 drinks for your 3 Red Cups. But let's say you are drinking a 6.75% alcohol beer (there are many of them, check here) you've just consumed ~7 alcohol drinks for your 3 Red Cups
The problem is compounded when hard alcohol is used. Let's say you make a mixed drink that is half hard alcohol / half juice in a Red Cup....
The drink you just made is not one standard drink!
In that one red cup, there are SIX drinks!
Stanford University - Office of Alcohol Policy and Education