Textbook Notes (270,000)

CA (160,000)

U of A (2,000)

STAT (100)

STAT151 (80)

Omar Rivasplata (20)

Chapter 12

This

**preview**shows half of the first page. to view the full**3 pages of the document.**CHAPTER 12: FROM RANDOMNESS TO

PROBABILITY

12.1: RANDOM PHENOMENA

Random Phenomena: A phenomenon is random is we know what outcomes could happen, but not which particular values will happen

In general, each occasion when we observe a random phenomenon is called a trial

oAt each trial, we note the value of the random phenomenon, and call that the outcome of the trial

Event: A collection of outcome; usually we identify events so that we can attach probabilities to them; we denote events with bold

capital letters, such as A, B, or C

Sample Space: The collection of all possible outcome values; the sample space has a probability of 1

A phenomenon consists of trials Each trial has an outcome Outcomes combine to make events

THE LAW OF LARGE NUMBERS

Law of Large Numbers (LNN): The relative frequency of an event in repeated independent trials gets closer and closer to its true or

long-run frequency as the number increases

oThis simpli4es things if we assume that events are independent

oIndependence (informally): Two events are independent if learning that one event occurs does not change the probability

that the other event occurs

For independent trials, one form of the LLN says that as the number of trials increases, the long-run relative frequency of repeated

events gets closer and closer to the single value

Since the LLN guarantees that relative frequencies settle down in the long run, we can now o5cially give a name to the value they

approach; it is the probability

Empirical Probability: The long-run relative frequency of an event’s occurrence

oThe law of large numbers assures us of its existence, when we can perform repeated trails

total

P

(

A

)

=¿×A occurs

¿of trials¿

THE NONEXISTENT LAW OF AVERAGES

The LLN says nothing about short-run behavior

oRelative frequencies even out only the in long run (which is in4nity long)

12.2: MODELLING PROBABILITY

Theoretical Probability: The probability that comes from a model (such as equally likely outcomes)

P

(

A

)

=¿of outcomes∈A

¿of possible outcomes

PERSONAL PROBABILITY

Personal/Subjective Probability: A probability is subjective and represents your personal degree of belief

THE FIRST THREE RULES FOR WORKING WITH PROBABILITY

1. Make a picture

2. Make a picture

3. Make a picture

###### You're Reading a Preview

Unlock to view full version