Probability: An Introduction with Examples and Applications, Lecture notes of Probability and Statistics

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Week 2: Probability

2 Outline

Week 2: Probability Example: Coin Tossing during WWII Why study Probability? What is Probability? Two types of Probability Describing Simple Probability Models Probability Results Random Variables Distributions Counting Short Cuts

2 Fortunately Kerrich was imprisoned in a camp in Jutland run by the Danish Government in a ‘truly admirable way’.

2 With a fellow internee Eric Christensen, Kerrich set up a sequence of experiments demonstrating the empirical validity of a number of fundamental laws of probability. I They tossed a coin 10,000 times and counted the number of heads. I They made 5000 draws from a container with 4 ping pong balls (2x2 di

erent brands), ’at the rate of 400 an hour, with - need it be stated - periods of rest between successive hours.’ I They investigated tosses of a ‘biased coin’, made from a wooden disk partly^ coated in lead. In 1946 Kerrich published his finding in a monograph, An Experimental Introduction to the Theory of Probability

.^

Kerrich1950Paper How many heads do you think he counted? What is the probability of getting a head on a fair coin?

2 Why study probability?

I Think back to the population-sample relationship and the 2016 Australian Federal^ Election example from last week. I Many complex processes (e.g. voting) are random, which means the final^ outcomes can vary greatly. Hence we might encounter great^ di

culties/inaccuracies when quantifying population behaviour using a sample. I Probability theory provides us with a mathematical framework to describe random^ outcomes with high degree of certainty. (E.g. most polling data do not use more^ than 1000 samples) I Thus, studying probability allows us to derive highly accurate results without the^ need to perform overcomplicated/costly experiments. By imposing a certain^ probability framework on our sample, (and if model assumptions are made^ correctly) then we can be quite certain that the trends we discovered in our^ sample can be replicated across the general population.

2 What is Probability?

Definition (Probability) The probability of an event is a measure of the likelihood of that event occurring.

Probability Theory is a set of mathematical tools which dates back centuries to^ casino type games. A more formal theory was developed in the 1930s by the^ Russian mathematician A. N. Kolmogorov. History Video Appendix

This is a good opportunity to tell you this very important fact: mathematics and English are two distinct languages

. The probability we are discussing here is very di

erent to what we mean by “chances” in everyday language. (e.g. “There is a one in six million chance that two jumbo jets will collide. ”)

2 Frequentist (or simulation) probability: based on data

The probability of an event is the proportion of times that event would occur in a large number of repeated experiments (simulation). In some ways, subjective probability is an informal version of frequentist probability, using one’s own history and experience (‘personal data’) as the basis. Tossing a Fair Coin The probability of tossing a fair head =

P

H

)^

= The long run proportion of heads if we toss a fair coin a large number of times.

2 Famous Coin Tossers Person Number of Tosses n Number of Heads x P^ (Head ) Count Bu ↵on (1707-1788) Bu ↵on 4040 2048

Karl Pearson (1857-1936) Pearson 24000 12012

John Kerrick (1903-1985 war camp) Kerrick 10000 5067

Perci Diaconis (1945-present) Diaconis machine

2 Kerrich’s Tosses: Proportion of Heads This illustrates the ‘Law of Averages’ or ‘Law of Large Numbers’: the proportion of heads becomes more stable as the length of the simulation increases and approaches a fixed number called the ‘relative frequency’.

2 Theoretical probability: based on model

The probability of an event is based on a model of the context. Two models for Coin Tossing: (1) Deterministic physics model: the side that the coin lands on is determined by a number of complicated factors such as which way up it started, the degree of spin, the speed and angle with which it left the thumb and how far it has to fall. PercisDiaconisVideo PercisDiaconisVideo

2

Extension Suppose you toss a fair coin 1000 times: when it lands heads you receive $1, and when it lands tails you pay me $1. What is your expected profit or loss? Using Simulation: set.seed

CoinTosses

sample

c(

,^1

replace

TRUE

plot

cumsum (CoinTosses), xlab ="Tosses"

ylab

="P/L"

type

"l"

0 200 400 600 800 1000 0 40 − Tosses P/L

2 We could run this simulation 10000 times, resulting in: Histogram of all the final P&Ls Frequency − 100 −^50 0 50 100 250 200 150 100 50 0

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