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Chapter 12

# STAT151 Chapter Notes - Chapter 12: And1, Mutual Exclusivity

Department
Statistics
Course Code
STAT151
Professor
Omar Rivasplata
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 outcomesA
¿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
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