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Probability of an event is its
longrun relative frequency
; Must be
legitimate
0 ≤ P ≤ 1 & sum of set of P’s = 1
Event
– combination of outcomes
For any random phenomenon, each attempt (or
trial
) generates an
outcome
(
Discrete
– distinct values /
Continuous
– some range)
“
Something Has to Happen Rule
” – the probability of the set of all possible outcomes of a trial = 1
P(S) = 1
S is set of all possible outcomes
Independent

Formal Def
 Events independent if P (BA) = P(B)
prob of independent events don’t change when you find out one of them has occurred
* No such thing as “law of averages” (promises shortrun compensation for recent deviations from expected behavior)
Law of Large Numbers
–longrun relative frequency of repeated independent events gets closer to true relative frequency as number of trials increases
Disjoint
(
Mutually Exclusive
) – events that can’t occur together; no outcomes in common
Complement Rule
– The probability of an event occurring is 1 minus the probability that it doesn’t occur
P(A) = 1 – P (A
C
)
Multiplication Rule
– for INDEPENDENT events A & B
“Some” means at least one
complement of this is none; so P(at least one A in 5 trials)
[ P(A) = .35 / P (A
C
) = .65;
P(none) in all 5 = (.65) ^ 5
Complement = 1 – ((.65) ^5)
Sample Space
– collection of all possible outcomes (which don’t have to be equally likely); Ex. If pulling bills from wallet
S = {1, 5, 10, 20, 50, 100}
General Addition Rule
– DO NOT need disjoint events;
General Multiplication Rule
– DO NOT need independence; P( A and B) = P (A) * P(B A)
OR
P(B) * P(AB)
Conditional Probability
– P ( B  A) = “prob of B given A”
*
Mutually Exclusive events CANNOT be Independent
Venn Diagram
:
Tree Diagram
:
Drawing w/o replacement
– denominator changes because one less available each drawing; Ex: drawing something randomly like rooms
**
P(BA) ≠ P(AB)
[Baye’s Rule reverses conditional probs]
Model for Distribution of Sample Proportions
How a
statistic
(phat, b1, ybar) is distributed,
NOT how data
(p
,
μ, β)
is distributed

Randomization leads to sampletosample variation
creates sampling distribution for samples of particular size n
Distribution is N (p, √(p*q / n))
Normal model
centered at true p with SD of √(p*q / n); Claim is more true as n grows…
Assumptions
: 1) Sampled values independent of each other 2) Sample size (n must be large enough)
Conditions
: 1)10 % Condition – sample size, n, must be no larger than 10% of population (if sample made w/o replacement)
2) Success/Failure Condition – the number of successes (n*p) and number of failures (n*q) must both be greater than 10
Central Limit Theorem
–the sampling distribution model of the
sample mean
&
proportion
is approximately Normal for large n,
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 Fall '07
 DRABICKI
 Macroeconomics

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