Notes From David Palay: Chapter 5.1 Introduction to Probability What are the chances that…

Probability From the book, – “The probability of an outcome is defined as the long- term proportion of times the outcome occurs.” From Wikipedia, – “Probability is a way of expressing knowledge or belief that an event will occur or has occurred.” Mr. David Palay, – “Probability is the chance something will or will not happen”

Terms Experiment – An activity where the outcome is uncertain NOT NECESSARILY UNKNOWN, JUST UNCERTAIN Outcome – Result of a single trial of an experiment Sample Space – Collection of all possible outcomes of an experiment Event – Collection of outcomes from the sample space of an experiment

Rules of Probability

Are these valid probabilities?

Rules of Probability (continued) For any given experiment, the probability of the sum of the outcome probabilities in the sample space must equal 1. – SOMETHING has to happen, or we have an incomplete sample space.

Experiment & Theory Experimental Probability: – Also called the “relative frequency method” – Probability we get from the results of running tests. Theoretical Probability: – Also called the “classical method” – The probability calculated based on the rules of mathematical probability. (Which we will touch on later)

Dice nomenclature

Some Examples ExperimentSample Space Example Events Roll 1d6 Flip two coins Randomly pick a billiard ball Rolling a six: {6} Rolling an even number: {2, 4, 6} Rolling under a 3: {1,2} Getting 2 heads {HH} Getting at least 1 head {HH, HT, TH} Picking a solid: {1, 2, 3, 4, 5, 6, 7, 8} Picking a yellow ball {1, 9} Picking the 8-ball {8}

Basic Probability

Ok, that sounds easy.. Find: P(rolling a 3 on 1d6): P(rolling odds on 1d6): Which is greater? Why?

More Practice Standard deck of cards: 4 suits {Spades, Diamonds, Hearts, Clubs} and 2-10, Ace, Jack, Queen, King. The Jack, Queen, and King are considered “Face cards” P(drawing a 3 from a shuffled deck): P(drawing a face-card of hearts):

Slightly harder now…

2d6 continued

So, we can see…

Law of Large Numbers Given a sufficiently large number (infinite) of trials, the Experimental Probability will approach the Theoretical Probability

The Great Glass Rod Problem If we take a glass rod, and break it at two random points, what is the probability that we will be able to form a triangle with the pieces.

Subjective Probability Intuition. Guessing. Personal Judgement.