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Lecture

# UASTAT141Ch14-15.pdf

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University of Alberta

Statistics

STAT141

Paul Cartledge

Winter

Description

Ch. 14 – Introducing Probability
Def’n: An experiment is a process that, when performed, results in one and only one of
many observations (or outcomes).
Probabilityis a numerical measure of likelihood that a specific outcome occurs.
3 Conceptual Approaches to Probability:
1) Classical probability
- equally likely outcomes exist when two or more outcomes have the same probability
of occurrence
- classical probability rule:
P(A) = (# of outcomes favourable to A) / (total # of outcomes for experiment)
2) Relative frequency concept of probability
- experiment repeated n times to simulate probability
- relative frequencies are NOT probabilities, they only approximate them.
- Law of Large Numbers: If an experiment is repeated again and again, the prob. of an
event obtained from the relative frequency approaches the actual or theoretical prob.
3) Personal (or subjective) probability
- personal probability is the degree of belief that an outcome will occur, based on the
available information
Calculating Probability
Def’n: A sample space (a.k.a. S) is the set of all elementary outcomes of an experiment.
evennt (a.k.a. A) is a subset of elementary outcomesA ⊂ S.
Æ P(A) = probability that A occurs
• A union of 2 events (A, B, or both happen) is denoted by A or B (or A∪ B).
• An intersection of 2 events (A and B happen together) is by A and B (or A∩ B).
C
• A complement of an event (event does not happen) is denoted by A .
A Venn diagram is a picture that depicts S (events above drawn in class).
Experiment Outcomes Samacele
Toss a coin Head, Tail S = { H, T }
Toss 2-headed coin Head S = { H }
Toss a $5 bill Get it back, Lose money S = { Lucky, Not Smart }
Pick a suit Spades, Clubs, Diamonds, S = { AceSp, 2Sp, 3Sp,…,
Hearts AceC, 2C,…, KingH }
(Associated Venn diagrams drawn in class)
Properties for calculating probabilities:
1. 0 ≤ P(A) ≤ 1
2. P(A) is the sum of probabilities of all elementary outcomes comprising A.
3. P(S) = 1 Ch. 15 – Probability Rules!
Basic Rules for Finding the Probability of a Pair of Events:
Table 15X0 – 2-way table of responses
Hockey Like Indifferent Dislike Hockey Total
(A) (B) (C)
Male (M)
Female (F)
Total
Def’n: Marginal probability is the probability of a single event without consideration of
any other event.
Ex15.1) P(M) = P(F) =
P(A) = P(B) = P(C) =
Condiptoonalbility is the probability that an event will occur given that
another event has already occurred. If A and B are 2 events, then the conditional
probability of A given B is written as P(A | B). Keywords: givenf, of
P(A| B) = P(A∩ B) and P(B | A) = P(B∩ A)
P(B) P(A)
such that P(A) ≠ 0 and P(B) ≠ 0.
Ex15.2) a) If you are male in this class, what is the probability that you like hockey?
P(A∩ M)
P(like hockey | male) = P(A| M) = P(M) =
b) What is the probability of being female in this class, given that you are indifferent to
hockey?
P(F ∩ B)
P(female | indifferent) =(F | B) = =
P(B)
Two events are independent if the occurrence of one does not affect the probability of the
occurrence of the other. In other words,
P(A | B) = P(A) OR P(B | A) = P(B)
Ex15.3) From Table 15X0, P(F) = P(F | B) =
Since probabilities are not equal, the 2 events are not independent.
Ex15.4) deck of cards: P(Black) = 26/52 = ½ P(Black | Face) = 6/12 = ½
Since the probabilities ARE equal, the 2 events are independent. Disjoint (or mutually exclusive) events are events that cannot occur together.
Ex15.5) deck of cards Ex15.6) a single die
R = get red suit Æ diamond or heart E = even = {2, 4, 6}
B = get black suit Æ spade or club O = odd = {1, 3, 5}
F = get face card Æ jack, queen, or king Pr = prime = {2, 3, 5}
Which pairs are disjoint?
Two important observations regarding disjoint, independent & dependent events:
1. Two events are either disjoint or independent, but not both (unless one has zero prob.).
2. Dependent events may be disjoint, but disjoint events are always dependent.
C
Complement Rule: P(A)C+ P(A ) = 1, so C
P(A) = 1 – P(A ) and P(A ) = 1 – P(A)
Ex15.7) From Table 15X0, P(Fem

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