Probability and Statistics 5

Probability and Statistics (Part 5)

By Madison Nef

Brownian Motion Theory
In the early 1800’s, many people didn’t understand the concept of atoms and molecules, dismissing them as metaphors for physical matter, since they can’t be seen. The breakthrough that finally convinced people was Einstein’s paper on Brownian motion- the theory invented in 1827 by Robert Brown. Brown noticed that grains of pollen had landed in water and were moving in spontaneous, jittery random motions on the water’s surface. The jitters were ALWAYS random and never slowed down and/or stopped. It was his belief that the atoms and molecules that made up the water and the pollen were hitting against each other, which was propelling the pollen and keeping it moving across the surface of the water.

Einstein invented a formula, using the temperature and velocity of the water as a gauge, and figured out (on average) how far a piece of pollen could go in any set amount of time. Einstein used this information to figure out that the pollen was displaced from its previous location at a rate that was proportional to the square root of time.

"God does not play dice with the universe." - Albert Einstein

Misunderstanding the weather

Surprisingly, one of the biggest misconceptions in the realm of probability is about something we use each and every day, without even knowing it: the weather! Many people misunderstand the weather report- if you see a notice online or on the news about “30% chance of rain”, what do you think it means? Before we answer that, two matters should be cleared up. 1: How much rain does there have to be for it to actually qualify as RAIN? The answer is 0.01 inches, or one hundredth of an inch. There must be at least that much rain measurable to count as actual rain. 2: What does it mean when there is a 30% chance of rain in one, specific spot? The answer to that is simple: all it means is that in weather circumstances like today, 30 out of 100 days you should expect rain in that spot.

The problem starts when there is supposed to be a 30% chance of rain in ONE REGION. Regions have many different points, so unless the region is homogeneous, it would be impossible to calculate the rain chance for the whole region… mainly because 50% of the region could have a 40% chance of rain while the OTHER 50% of the region could have a 20% chance of rain. To accurately calculate the chance of rain for the whole region, we simply average out the two percentages and get 30% chance of rain for the whole region.

Below is the definition for the probability of precipitation as described by the National Weather Service… it’s not very helpful, but it is better than nothing:

The "Probability of Precipitation" (PoP) describes the chance of precipitation occurring at any point you select in the area.
How do forecasters arrive at this value? Mathematically, PoP is defined as follows:
PoP = C x A where "C" = the confidence that precipitation will occur somewhere in the forecast area, and where "A" = the percent of the area that will receive measureable precipitation, if it occurs at all.
So... in the case of the forecast above, if the forecaster knows precipitation is sure to occur (confidence is 100%), he/she is expressing how much of the area will receive measurable rain. (PoP = "C" x "A" or "1" times ".4" which equals .4 or 40%.)
But, most of the time, the forecaster is expressing a combination of degree of confidence and areal coverage. If the forecaster is only 50% sure that precipitation will occur, and expects that, if it does occur, it will produce measurable rain over about 80 percent of the area, the PoP (chance of rain) is 40%. ( PoP = .5 x .8 which equals .4 or 40%. )

In either event, the correct way to interpret the forecast is: there is a 40 percent chance that rain will occur at any given point in the area.