Basic Probability

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Basic Probability

• The laws of chance

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overview

• a) discuss laws of probability, which are useful
• b) define combinations (and permutations)
• c) use a) and b) to develop the binomial
  distribution, which is useful.

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experimental roots

• motivation game theory gambling
• apparatuses coins dice
• playing therewith leads to notions of
• the frequency, m, of an event
• the relative frequency, m/n, of an event

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basic single die experimenthow many times, m,
will 4 (say) come up?

• intuition
• die has a symmetry suggesting all faces are
  equally likely to come up
• definition based on intuition
• the chance or probability that an event will
  occur is

(equivalent to analytical view in Howell)
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a priori probability

• let a be an event that can occur in A ways
• let B be the number of ways that anything else
  can occur
• then

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assumptions we just made

• the truth of the (unique) case that all ways
  are equally probable.
• In other words, we assumed a null hypothesis.
• (how would we test it for a given die?)
• successive throws are totally independent
• In other words, the die has no memory.
• (which often violates our intuition)

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results of the experiment
success was a roll of 4
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(which leads to)an alternate definition

• which is the
• a posteriori probability
• or the relative frequency view from Howell

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Two important theorems

• Theorem of total probability
• a.k.a. the additive law
• Theorem of joint probability
• a.k.a. the multiplicative law

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Theorem of Total Probability
If A and B are independent events, then the
probability of A or B occurring is equal to the
sum of the probabilities of A and B. In other
words
(assuming A and B are mutually exclusive)
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Theorem of Joint Probability
If A and B are independent events, then the
probability of A and B occurring is equal to the
product of the probabilities of A and B. In
other words
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Conditional Probabilityand the full Theorem of
Joint Probability
The probability of A and B occurring is equal to
the product of one probability and its
conditional probability given the the other. In
other words
(which is just p(A)p(B) if A,B are independent)
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Venn Diagrams and this stuff
mutually exclusive
not mutually exclusive
p(B)
p(A)
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more general additive law
not mutually exclusive
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probability example two coin flips
venn diagram
outcome matrix
toss two
0
1
0
toss one
1
A toss 1 is heads
both are heads
B toss 2 is heads
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example calculations
p(A) p(B) 1/2
A
B
AB
p(A and B) intersection 1/2 1/2 1/4
toss two
0
1
p(A or B or both) union 1/2 1/2 - 1/4 3/4
0
toss one
(by the general additive law)
1
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Combinations
is the number of ways (in any order) to get m
heads by flipping n coins, and is given by

• (I learned so you might see it written this
  way too)
• (also number of combinations of n things taken
  m at a time. How can this be? What are the
  things?)
• (Permutations order matters, omit m!)

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The Binomial Distribution

• Which we can break down as follows
• a p(head), b p(tail)
• second two termsp(one a and another and a b,
  and another) is just a joint probability
• the first term is the number of equally-likely
  ways this could happen, so the probability of any
  of them happening is is the sum of all C of the
  probabilities (additive law)

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simple binomial distribution
p(x heads) on one coin flip
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binomial sample size
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The Gaussian Approximation
mean na var nab
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binomial p(heads)
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Usefulness of the Binomial

• confidence limits on, for example, opinion polls
  (the margin of error).
• by extension, hypothesis testing (was our die
  fair?)
• the sign test (great for ordinal variables)

results