Here are some ball park numbers :
1> Volume of a typical fridge = 20 ft3 or 0.6 m3 (trying to use round numbers
" When it comes to the interior, the refrigerator itself should provide you with 14 to 20 cubic feet of space. " What's the Standard Refrigerator Size? - THOR Kitchen.
2> Volume of a beer can = 12 oz or 355 ml or 0.0004 m3
3> For estimation purposes, lets assume the fridge is filled with beer cans. With packing efficiency and space - lets assume that 70% of the fridge volume is filled with beer cans. That gives : 0.6 x 0.7 / 0.0004 ~ 1000 cans of beer
4> The surface area of a 12 oz beer can is 67 in2 What Is the Surface Are of a Soda Can? So 1000 beer cans will have a surface area of 67 x 1000 = 67,000 in2 = 43 m2 (Not all surface area will be exposed but for our case - lets assume it is )
5> Lets assume the beer is cooled inside the fridge to 50 F (10 C) and the room temperature is 75 F (24 C)
6> Heat transfer coefficient in air is typically about 10 W/m2/K Convective Heat Transfer Coefficients Table Chart
7> So the heat lost over 3 mins (180 seconds) is :
10 x 43 x (24 - 10) x 180 = 1083600 Joules
8> Coefficient of performance of a refrigerator is about 3 to 4. Lets assume 4 https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.438.9480&rep=rep1&type=pdf This means 1J of energy is consumed per 4 joules of energy removed.
9> So the electrical energy consumed over the 3 minute period is 1083600/4 = 270900 J = 0.08 kWh
10> 1 kWh in the US is about 10c. So 0.08kWh is about 1c. So you waste about 1c in electricity if you leave the fridge open for 3 mins.
11> Do not use my calculations to convince your parents or spouse. It doesn’t work.