Recall October 29, 2012: High winds during Superstorm Sandy; record-breaking high tides, mandatory evacuations, significant power outages, $1Million in estimated property damage.

Recall February 24, 2016: Strong winds; a tractor blown over on the upper level of the George Washington Bridge, a sidewalk shed collapsed on Lenox Avenue in Harlem, $100k in estimated property damage.

Recall August 18, 2009: Thunderstorm winds; hundred trees down in Central Park, significant tree damage in western Central Park between 90th and 100th Street, $300k in estimated property damage.

Recall conditional probability rule from lesson 9.

P(A|B) = P(A ∩ B) / P(B) or P(A ∩ B) = P(A|B)*P(B)

Recall mutually exclusive and collectively exhaustive events from lesson 4.

The midterms are mutually exclusive, but the final exam is collectively exhaustive.

If you have *n* events ** E1**,

**, …**

*E2***that are mutually exclusive and collectively exhaustive, and another event**

*En***that may intersect these events, then, the**

*A***law of total probability**says that the

*probability of A is the sum of the probabilities of its disjoint parts*.

Let us cut the jargon and try to understand this law using a simple example. We have the **data on property damage during storms** accessible from NOAA’s storm events database. Let us take a subset of this data — wind storms in New York City. You can get this subset here.

With some screening, you will see that there are 57 events → 16 high wind events, 15 strong wind events, and 26 thunderstorm wind events. Notice that there is property damage during some of these incidents. Let us visualize this set up using a Venn diagram.

Your immediate perception after seeing this picture would have been that the high winds, strong winds, and thunderstorm winds are mutually exclusive and collectively exhaustive. They don’t intersect, and together make up the entire wind storm sample space. **Damages cut across these events.**

Let us first focus on the high wind events. The 16 high wind events are shown as 16 points in the picture below. Notice that 4 of these points are within the damage zone. The probability of high wind events is P(H) = 16/57, the probability of high wind events and damage is P(damage ∩ high winds) = 4/57 and the probability of damage given high wind events is

P(damage|high winds) = P(damage ∩ high winds) / P(high winds) = 4/16

Now let us add all the other points (events) onto the picture. Some of these will be in damage zone, and some of them will be out of damage zone.

We can estimate the total probability of damage by adding its disjoint parts.

P(damage) = P(damage ∩ high winds) + P(damage ∩ strong winds) + P(damage ∩ thunderstorm winds) or P(damage) = P(damage|high winds)*P(high winds) + P(damage|strong winds)*P(strong winds) + P(damage|thunderstorm winds)*P(thunderstorm winds) P(damage) = (4/16)*(16/57) + (8/15)*(15/57) + (9/26)*(26/57) = 21/57

The best part is that we can use this law as a **predictive equation**. Suppose there is an approaching storm and the weatherman told you that there is a **10%** chance that the coming storm has high winds, **30%** chance that it has strong winds and **60%** chance that it has thunderstorm winds, you can immediately use this law and compute the probability of damage for NYC.

Can you tell me what that damage probability is?

Should I wait till after your Earth Day March?

Recall that you are totally contributing your share of Co2 to the earth during the March.

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