Chapter 3 – Probability

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Introduction to Probability for Statistics
Introduction to Probability for Statistics
Recall that we are studying two types of statistics – descriptive statistics and inferential
statistics. In order for us to move to inferential statistics, we need a certain level of
understanding of probability. In this section, we will study the basics of probability
needed to study inferential statistics later. This section covers the basics of probability in
three subsections: fundamentals, addition rule, and multiplication rule – basic.
Event – collection of results or outcomes of procedure.
Simple event – event or outcome that cannot be broken down (simplified) any more.
Sample space – all possible simple events or outcomes.
Rule 1: Relative Frequency Approach (based on past experience)
P( A) 
number of occurances of A number of successes
total number of trials
total number of trials
Rule 2: Classical Approach (equally likely outcomes)
P( A) 
number of ways Acan occur
total number of sample events n
Rule 3: Subjective Approach – estimating or guessing P(A) based on
knowledge of circumstances.
Unusual event – If P(A)  0.05, then event A is said to be an unusual event.
Law of Large Numbers – as the number of trials increase, the relative frequency
probability approaches the actual probability.
Complement of event A is A equals all outcomes not containing event A.
Rounding answers: exact fraction or decimal of three significant digits. 0  P(A)  1.
P( A) a
 (in reduced form). a : b or a to b.
P( A) b
P ( A) b
 (in reduced form) b : a or b to a.
Odds in favor of event A =
P ( A) a
net profit
Pay off odds against event A =
amount bet
Odds against event A =
Introduction to Probability for Statistics
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Addition Rule
Compound event – two or more simple events.
Addition rule: P(A or B) = P(A) + P(B) – P(A and B)
Mutually exclusive events A and B cannot occur simultaneously.
If A and B are mutually exclusive, then A and B have nothing in common
and P(A or B) = P(A) + P(B) because P(A and B) = 0.
Rule of Complementary Events: If A and A are complementary events, then
P(A) + P( A ) = 1 or P(A) = 1 – P( A ) or P A ) = 1 – P(A)
Multiplication Rule - Basic
P(A and B) = P(event A occurs and event B occurs)
Conditional Probability:
P(B|A) = P(event B occurs after event A has already occurred)
Independent events: occurrence of one does not affect the probability of the
occurrence of the other event.
P(A and B) = P(A)P(B|A) general rule
If A and B are independent, P(A and B) = P(A)P(B) because P(B|A) = P(B).
Common guideline for use of independent vs. dependent:
If a sample size is  5% of population size, treat the selections as being
independent (even if selections are without replacement; they are technically