Hello all! For the 4th Math Monday, I have decided to do two quick proofs to show you how to write a proof.
“How can we measure the probability of drawing something without replacement?”
Sometimes, we would like to know the probability of choosing certain types of cards and that we either choose the right or wrong card. The problem with this is that the probability will change with each card drawn. As a result, this will take on a special Hypergeometric Distribution.
The probability of choosing the desired card can be found with the equation
With N being the total population, n being the number of draws, K being the number of successful members of the population, and k being the amount of the successful members desired.
Image credit: Wikipedia
Welcome to the 3rd installment of Matt’s Math Mondays! I am pleased to introduce to you one of the most powerful and applicable ideas in math known as groups! I hope you all have as much fun reading it as I had writing it.
On Convergence of Infinite Sums
“What is so special about infinite sums coming to a specific value?”
We have all seen the classic sum.
Which is often represented as…
We, in our high school careers, have been taught that this sum is 2. This is
most often done so with an algebraic manipulation of both sides of the sum. We
haven’t been taught how to verify this sum. But how would we go about doing
The basic idea is that if I end my sum at nite point, my result will deviate
from the actual sum by some error (which we will call ε. This error is some
positive real number. To verify that this sum indeed approaches 2, it suffices
to show that for any error, there exists a nite point in the series in which the
difference between 2 and this point is less than the error AND all further points
will also be less than the error.
Probabilistic Risk Assessment
“How can we determine the risk of an infrastructure project?”
Grand infrastructure projects such as nuclear power plants, high-speed rail, and hydroelectric dams are essential for the backbone of modern human civilization. However, there if there are any faults in these systems, then the consequences could be very severe. As a result, these projects have to undergo Probabilistic Risk Assessment, which entails determining the magnitude of possible consequences and their likelihood suing quantitative analysis.
I = PAT
“How can we represent environmental impact based on resource consumption?”
Every society has an environmental impact. But since each society has different levels of resource consumption and utilization, each one is different. So how can we quantify this? Well, we can measure impact based on the average of the amount used multiplied by its energy efficiency and the total population. This is known as the I = PAT equation where I is the Environmental Impact P is the Total Population A is the Level of Affluence (how many resources are used) and T is the Technological Efficiency.