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Basic Probability

- The laws of chance

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.

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

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)

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

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)

results of the experiment

success was a roll of 4

(which leads to)an alternate definition

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

Two important theorems

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

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)

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

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)

Venn Diagrams and this stuff

mutually exclusive

not mutually exclusive

p(B)

p(A)

more general additive law

not mutually exclusive

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

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

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!)

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)

simple binomial distribution

p(x heads) on one coin flip

binomial sample size

The Gaussian Approximation

mean na var nab

binomial p(heads)

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

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