Help with probability?

shizle

I am writing a paper for a science essay competition on the subject of probability. We have have yet to study the LC HL maths probability section, so I studied it on my own to learn how to do basic probability calculations which I could use in my essay. I know how to calculate odds of lotto jackpots, the birthday problem, etc.

I am having trouble finding simple information about how probability theory is used in different industries, e.g. in stock markets and insurance. Would anyone be able to explain to me in basic language how probability is applied in circumstances such as this? Obviously the technicalities would be far too complicated for me to grasp, however in broad terms, how is it used?

Also, any other suggestions for possible topics to cover in my essay, other than calculating basic odds for lotteries, use of probability in gambling and in insurance and stock markets?

I'll have a go at helping out. Uses are in bold and jargon is in italics. Here it goes!

In insurance and stock market industries, probability is used when doing 'risk assessment'. This basically means that the insurance companies try to actually put a number (quantify) on the risk that they take by insuring a particular house, car, laptop, whatever it may be. To do this they use probability.

The risk associated with insuring a house might be related to the probability of it being robbed. Obviously in this case they cannot work out exactly what this probability would be, but they can use things like the local crime rates to estimate it.

Using what are called Probability Distributions, in which Probability plays an integral role, far more applications open up. These are vital for conducting scientific or social experiments.

For example, suppose a researcher did a survey on hours spent studying by Leaving Certificate students in 2010. The researcher does not have the time or the money (or patience ¬_¬) to give this survey to every student who was doing the Leaving Cert. Instead, he/she may decide to give it to 1000 students across the country and find the average time spent by these students studying.
To find out if this average is close to the average for every student in the country, he/she can use Probability and Probability Distributions.

Other areas include Weather Forecasting and Physics.

Weather forecasts may tell you there is a 50% chance of rain. How on Earth do they work that one out?
Well, they look back through their database for days with similar conditions (judging what 'similar conditions' are mathematically is actually quite hard, as an aside) and find the proportion of days like this that it rained. This is the number of days that it rained divided by the total number of days, and it is the same as the probability of rain on a day like this.

In Physics, Probability is becoming more important, as when you have large systems with a lot of different objects interacting with each other, we sometimes can't write down a simple formula to predict how the system will change after some time... we are resigned to talking about the probability of a system changing this in way. (This is called a non-deterministic model).

You mentioned the Birthday Problem. The other famous problem in Probability is the Monty Hall problem. Like the Birthday Problem, it surprises most people upon learning the answer.
Monty Hall Problem Wikipedia Page

This is just a taste of some of the applications I know about, but there are some good websites out there. What would a decent post be without good links, huh?

• Some basic examples of probability in everyday life.
• Odd things that probability tells us.
• A whole blog about probability in the real world. It's actually really a tutorial about probability, not that interesting. If you haven't covered probability though, might be worth taking a look at it.

And here's a very interesting video; there's a few more like it on youtube.

https://www.youtube.com/watch?v=U9-G-noZrwc

Fremen

You can use probabilistic methods to prove non-probabilistic theorems. For instance, there is a proof of the fundamental theorem of algebra (that polynomials with complex coefficients have complex roots) in terms of Brownian motion in the plane.

http://en.wikipedia.org/wiki/Probabilistic_proofs_of_non-probabilistic_theorems

You need to learn about the central limit theorem if you haven't already. It's the cornerstone of probability theory, and explains why the normal distribution is called "normal".

Edit: communications theory uses a whole lot of probability. A signal is modelled as a "stochastic process" (look it up). It has a signal component and a noise component. The noise component is inherently random. There are laws which quantify how much information you can carry across a noisy channel (the shannon information capacity).

Probability is also used in target tracking. If the Russians launch a bomber at the USA, and the USA has a number of tracking stations along the flightpath, they can make a series of measurements of the position of the bomber which may contain errors. There is an algorithm called the Kalman filter which tells the Americans what the best guess of the position of the bomber is.

Edit #2: It's quite interesting that as a mathematical discipline, probability has been around for a few hundred years: at least since the Bernoullis. However, it was only made mathematically rigorous by Kolmogorov at the beginning of the 20th century. Unless you're focusing entirely on applications, you could mention a little about the modern history of probability.

Fremen

Take a look at monte-carlo methods, too. The monte-carlo method is a ridiculously useful way of computing numerical approximations.
The wikipedia article gives a fairly user-friendly introduction.

shizle

Thanks a million everyone!:D Lots of really interesting information and very easy to understand. I've entered my essay now anyway, so we'll see what happens!