Probability

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Probability
Sociology 315, Winter 2002 Week 8 Feb. 25 -
March 1
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Approaches to Probability
Classical probability based on gambling ideas,
the fundamental assumption is that the game is
fair and all elementary outcomes have the same
probability the composition of the population
is known - think of a card game, or a dice.
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Empirical (relative) probability when an
experiment can be repeated, then an outcomes
probability is the proportion of times the
outcome occurs in the long run - empirical
probabilities are based on actual measurements,
not theory not common in the social
sciences. Personal (subjective) probability
many of lifes outcomes are not repeatable.
Personal probability is an individuals personal
assessment of an outcomes likelihood - based
on beliefs or hunches.
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Events vs. Outcomes
Possible Outcomes 1 Getting heads 2 Getting
tails 1 Getting a spade 2 Getting a heart 3
Getting a diamond 4 Getting a club 1 Failing 2
Passing
Event Tossing a coin Choosing a playing
card Taking an examination
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Formal Properties of Probability
1. Probabilities range between 0 and 1.
0 ? p (A) ? 1 0 is less than or equal to p(A)
which is less than or equal to 1 Probability
is often expressed as a proportion or as an odds
p (A) .28
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2. Addition Rules a. When adding two mutually
exclusive outcomes. p (A) p (B) p (A or
B) or p (A U B) Uunion What is
the probability of outcome A and outcome B
occurring together p(A and B)? Zero, when
mutually exclusive.
A
B
Mutually exclusive Two outcomes that do not
occur at the same time (no overlap).
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If we want to know the probability of A OR
B occurring, we simply add their respective
probabilities together.
p (A) p (B) p (A or B) or p (A U
B) Likewise, p (A) p (B) p (C) p (A or B
or C) or p (A U B U C)
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b. Adding two outcomes which are not mutually
exclusive (can occur at the same time) p (A
or B) p (A) p (B) p (A and B)
? intersection and
A
B
p (A or B) p (A) p (B) p (A and B) The
intersection of A and B must be removed.
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Event Tossing a die Outcome Getting an
even score or getting a score greater than
3 Even numbers 2 4 6 Number greater than
3 4 5 6
4
2
5
6
1
3
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P(E) 3/6 0.5 P(G) 3/6 0.5 If these were
mutually exclusive outcomes P(E) P(G) 3/6
3/6 .5 .5 1 Because there is
overlap, we must subtract the intersection of E
and G P(E) P(G) - P(E and G)3/6 3/6 - 2/6
4/6 or .5 .5 - .33 .66
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3. Multiplication Rules a. Multiplying two
independent outcomes (the occurrence of A has no
effect on B) the proba- bility of the occurrence
of two outcomes is the product of the separate
probabilities of each outcome. p(A intersection
B) p (A) p (B) p (AB) The probability of A
and B together is the joint probability, or the
intersection of A and B.
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Joint Probability The probability of two
outcomes occurring simultaneously. What is the
probability of a respondent being high social
class and reform?
Party
High SES (S)
L.SES (notS)
TOTAL
Liberal (L)
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19
24
Reform (not L)
10
6
16
15
25
40
TOTAL
P (S and notL) 6/40 .15
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Marginal Probability The probability that
reflects independent outcomes. What is the
probability of a respondent being liberal?
Party
High SES (S)
L.SES (notS)
TOTAL
Liberal (L)
5
19
24
Reform (not L)
10
6
16
15
25
40
TOTAL
p (L) 24/40 . 6
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What is the probability of drawing a heart and
a queen? A drawing a heart
p(A)1/4.25 B drawing a queen
p(B)1/13.077 Joint Probability p (B)
.077 .019 prob. of a heart p (A) .25 p (A) p
(B) and a queen p (B) .923 .231 prob. of a
heart, p (A) p (B) not a queen p
(B) .077 .058 prob. of not p (A) .75 p (A)
p (B) a heart and a queen p (B) .923 .692
prob. of not a p (A) p (B) heart not a queen
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Other (not B)
Queen (B)
TOTAL
Heart (A)
.01925
.231
.25
Other Suit (not A)
.05775
.69225
.75
.077
.923
1.00
TOTAL
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b. Outcomes which are not independent (i.e. are
dependent) - when the occurrence of a given
outcome affects the probability of the
occurrence of another outcome - the
conditional probability rule must be applied.
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4. Conditional probability rule p (B A) is
the probability of B occurring, given that A has
already occurred. When A and B are not
independent p (A and B) p (A) p (B A)
therefore, p (B A) p (A and B)
p (A) (15/100) / (20/100)
.75 .15/.20 .75 When events A and B are
independent, then P (A B) p (A) it doesnt
matter that B has occurred.
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Probabilities and continuous variables In order
to determine the probabilities for continuous
variables, we must use the standard normal
curve. 1. We must first know the mean and the
stand. deviation from which a score was drawn. 2.
Calculate the z-score. 3. Using a table which
provides the proportions of area under the curve,
calculate the probability of that particular
score.
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Suppose that a college entrance exam has a mean
of 300 and a standard deviation of 60. What is
the probability that an individual selected at
random will score 150 or higher? z 150-300/60
-2.5 From Column C, Table A Area beyond z
.0062 There is a .9938 probability that an
individual selected at random will have a score
of 150 or more. (The probability of selecting,
at random,a person with a score of 150 or less
is .0062
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One and Two-Tailed p Values Refers to tails of
the normal curve. One-tailed when we are
interested in one end of the distribution. Two-ta
iled when we are interested in comparing
situations, i.e. as rare as or as common
as