Question

Suppose we have produced a machine that automatically fills beverage cups to serve at a restaurant. The volume of the cups are precisely 450 cc. The machine takes each cup and fills it with the following: ice cubes, liquid A, liquid B, and liquid C. Liquids A, B, C when combined properly make for an amazing drink that Professor Thomson loves. On average, the machine will dispense 40 cc total volume of ice, 150 cc of liquid A, 210 cc of liquid B, and 50 cc of liquid C. Since you learned from physics class that 20% of ice remains above the surface of water (80% below the surface) you’ve come up with he following equation for the total volume of the drink per cup:

V_total = V_A + V_B + V_C + 0.8 × V_ice

where V_ice is the total volume of ice .

Assume the following is know about the standard deviations of volumes:

σ_A = 2.5 cc

σ_B = 3.0 cc

σ_C = 2.0 cc

σ_ice = 12.5 cc

Determine the probability that a cup will overflow when filled.

Answer 1

E(V total) =E(VA+VB+VC+0.8*VIce)=E(VA)+E(VB)+E(VC)+.8*E(VIce)=150+50+210+0.8*40=442

V(V total)=V(VA)+V(VB)+V(VC)+.8²*V(VIce)=119.25

S.D(V total)==10.92

we assume that Filling of each liquid happens according to normal probability law and are independant. Hence the V total will have normal distribution with mean 442 and standard deviation 10.92

Required probability is

Hence the cup will overflow 23.21% of the times