# Probability and its Limits

## Subject:

## Overview

The modern theory of probability is considered to have begun in 1654 with an exchange of letters between Blaise Pascal and Pierre de Fermat, and has developed since then into the discipline which examines uncertain processes. For example, although on tossing a coin you have no idea whether you will obtain heads or tails we know that if you keep doing it then in the long run it is very likely that the proportion of heads will be close to a half. The lecture will discuss this and other examples of random processes e.g. random walks and Brownian motion.

## Extra lecture materials

## Transcript of the lecture

**PROBABILITY AND ITS LIMITS**

**Slide: Title slide**

**Slide: Overview of the lecture**

*Sample spaces and probability*

*Fermat and Pascal*

*Taking it to the limit*

*Random walks and bad luck*

*Einstein and Brownian motion*

**Slide: Experiments and Sample spaces**

*sample space*.

**Slide: Equally likely**

**Slide: Equally likely with events**

*event*is the name for a collection of outcomes.

**Slide: Founders of Modern Probability**

**Slide: A gambling problem: the interrupted game**

**Slide: Interrupted game: Fermat’s approach**

**Slide: Interrupted game: Pascal’s triangle**

*Arithmetical triangle*had been used in various cultures for more than 500 years.

*principle of mathematical*induction, and, in fact, gave the first explicit statement of this principle of mathematical induction.

**Slide: Interrupted game: Pascal’s approach illustrating Fermat**

**Slide: Interrupted game: Pascal’s approach of first to 5**

**Slide: Toss coin 10 times**

**Slide: Law of large numbers different values of n**

*n*increases the distribution gets more and more concentrated around 0.5

**Slide: Symmetric random walk**

**Slide: Coin Tossing**

**Slide: Ten Paths**

**Slide: Law of long leads or arcsine law**

*one case out of five*the path stays for about

*97.6%*of the time on the same side of the axis.

**Slide: one case out of ten**

*one case out of ten*the path stays for about

*99.4%*on the same side of the axis.

**Slide: coin tossed once per second**

*one in twenty cases*the more fortunate player is in the lead for

*364 days 10 hours.*

*one in a hundred cases*the more fortunate player is in the lead for

*all but 30 minutes.*

**Slide: Number of ties or crossing the horizontal axis**

*four days*to produce, on average,

*four times*as many ties or crossings as a

*one*-day game

*doubles*, that is, on average, increases as the

*square root*of the time.

**Slide: Antony Gormley's**

*Quantum Cloud**Quantum Cloud*. It was commissioned for a site next to the Millennium Dome in London. At 30m high, I believe it is Gormley's tallest sculpture to date (taller than the

*Angel of the North*). It is constructed from a collection of tetrahedral units made from 1.5m long sections of steel. The steel section were arranged using a computer model using a random walk algorithm starting from points on the surface of an enlarged figure based on Gormley's body that forms a residual outline at the centre of the sculpture

**Slide: Ballot Theorem**

**Slide: Simulation of Brownian motion**

**Slide: From random walks to Brownian motion**