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

• We write the probability of an event E as

–Which means that the probability of any event is between 0 and 1. • 0 means it will NEVER EVER EVER EVER HAPPEN.• 1 means it will ALWAYS happen

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

d – read “x dee y”, represents throwing x fair dice, each with y sides.e.g., • 3d6 (“three dee six”) represents rolling 3 six

sided dice.• 1d20: 1 twenty sided die

Some ExamplesExperiment Sample

SpaceExample 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

P (E )=number of ways Ecanhappentotal possible outcomes

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…

• What single sum has the highest probability of coming up when we roll 2d6?

• We need to figure out how many possibilities there are.– Ah HA! Counting! We have 2 “spots”, each with 6

possibilities. So….

2d6 continued1 2 3 4 5 6

1

2

3

4

5

6

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.