Why does relative humidity drop if I open the window in winter? What about the temperature drop? I understand that 40% rel. hum. at 0 °C is less absolute humidity than 40% at 20 °C. However, assume that we have 2$~m^3$ of air. Both are 40% humid at 20 °C. Now we exchange 1$~m^3$ with air that is 40% humid at 0°C. Then according to this picture we should get 7$~g\, m^{-3}$ and 2$~g\, m^{-3}$ of absolute humidity for each part. So that's 4.5$~g\, m^{-3}$ on average. Now the temperature should be 10 °C (right?). Then, 4.5$~g\, m^{-3}$ means a relative humidity of around 50%! So why does relative humidity drop in reality? Is it because the temperature would actually be higher somehow?
Let's do this with accurate equations. We know how to calculate the saturated water vapour pressure for temperature $T$:
$$e_s(T)=e_{s0}e^{\frac{h_i}{R_w}(\frac{1}{T_{s0}}-\frac{1}{T})}=610\text{ Pa} \times e^{5423(\frac{1}{273\text{ K}}-\frac{1}{T})}$$
$e_s$ is saturated water vapour pressure as function of $T$ in Pascals
$T$ is the temperature in Kelvins
So, the saturated water vapour pressure at $T=0°C=273K$ is equal to $e_s(273K)=610 Pa$. Similarly, $e_s(293K)=2360Pa$ and $e_s(283K)=1230Pa$.
So we have two parcels of air: 1. 20°C, 40% and 2. 0°C, 40%. What are the masses of the water vapour in both parcels?
$$e=f\times e_s(T)$$
$f$ is relative humidity, $e$ is the water vapour pressure
$$m(T)=\frac{eV}{R_w T}=\frac{f\times e_s(T)\times V}{R_w\times T}=\frac{fe_s(T)V}{461\frac{J}{kg}T}$$
$$m(293K)=0.0070kg$$
$$m(273K)=0.0019kg$$
The next step is to combine these two masses:
$$M=m(293K)+m(273K)=0.0089kg$$
So the density is $\rho=\frac{M}{V}=0.0045\frac{kg}{m^3}$
The water vapour pressure is given by:
$$e=\rho RT=587Pa$$
The relative humidity is then:
$$f=\frac{e}{e_s(283K)}=0.477$$
So this is around 48% of the relative humidity.
Why did I put here so many equations? Simple! These equations hold on high degree of accuracy, so they can't be wrong. Why would the relative humidity drop in reality?

Also, for Skew-T Log-P lovers:

This solution is even more concise and less error prone.