2.3 Weather

First Solution - Independence

If we assume that Sunday and Saturday weather are independent then the sum-rule (Section 1.5) applies:

$$\begin{array}{c}P(\text{rain Saturday }\mathbf{\text{or}}\text{ rain Sunday})=\\ P(\text{rain Saturday})+P(\text{rain Sunday})-P(\text{rain Saturday }\mathbf{\text{and}}\text{ rain Sunday})\\ =P(\text{rain Saturday})+P(\text{rain Sunday})-P(\text{rain Saturday})\times P(\text{rain Sunday})\\ =0.25+0.25-0.25\times 0.25=0.4375& \text{(2.1)}\end{array}$$

The diagrams in Figure 1.3 are useful in making this calculation more intuitive, especially the term where we subtract $$P(\text{rain Saturday})\times P(\text{rain Sunday})$$ because otherwise we over count the double-rain weekends.

Second Solution - Correlation

Is it really reasonable that rain on Saturday and Sunday are independent events? Probably not! It's probably the case that knowing that it rained on Saturday, then rain on Sunday is more likely. It may also be that if it didn't rain on Saturday then it will be less likely for rain on Sunday. So we'd have information possibly like:

$$\begin{array}{c}P(\text{rain Sunday}|\text{rain Saturday})=0.35\\ P(\text{rain Sunday}|\text{not rain Saturday})=0.15\end{array}$$

Knowing this changes the equation as

$$\begin{array}{c}P(\text{rain Saturday }\mathbf{\text{or}}\text{ rain Sunday})=\\ =P(\text{rain Saturday})+P(\text{rain Sunday})-P(\text{rain Saturday }\mathbf{\text{and}}\text{ rain Sunday})\end{array}$$

Notice, however, that we don't have a direct expression for $P(\text{rain Sunday})$ anymore. We only have the conditional or dependent forms, like $P(\text{rain Sunday}|\text{rain Saturday})$. We can use the marginalization procedure (Equation 1.13 on page 47), and sum over all of the conditional expressions

$$\begin{array}{c}P\left(\text{rain Sunday}\right)=P(\text{rain Sunday}|\text{rain Saturday})P\left(\text{rain Saturday}\right)+\\ P(\text{rain Sunday}|\mathbf{\text{not}}\text{ rain Saturday})P\left(\mathbf{\text{not}}\text{ rain Saturday}\right)\\ =0.35\times 0.25+0.15\times (1-0.25)=0.2\end{array}$$

and then we have

$$\begin{array}{c}=P(\text{rain Saturday})+P(\text{rain Sunday})-P(\text{rain Saturday})\times P(\text{rain Sunday}|\text{rain Saturday})\\ =0.25+0.2-0.25\times 0.35=0.3625& \text{(2.2)}\end{array}$$

which makes it less likely to rain on the weekend if the Sunday rain is correlated with the Saturday rain (Equation 2.2) than if they are independent (Equation 2.1). Why is that?

One way to think of it is that, although the probability of rain on Sunday is increased due to rain on Saturday, it is more likely that Saturday is not rainy. In those cases, which are more frequent, Sunday is less likely to be rainy as well. When the two days are independent, Sunday's rain is the same probability regardless of Saturday's weather. When they are dependent, then the more often clear Saturday weather makes it a little less likely for the Sunday rain, and thus lowers the chance of weekend rain by a little bit.

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.