4. Properties of pmf
• Basic (as defined by book)
• Important derived properties
(T 1.2-1 - T1.2-6)
• Strategies of T1.2-3 and T1.2-5
particularly important
5. Example
• (Ex 1.2-1 in book)
• Draw a card at random. Define events:
• A = {x: x is face card}
• C = {x: x is a club}
• Calculate:
• P(A), P(C), P(A ∩ C), P(A ∪ C)
• P(A’ ∪ C’)
7. Motivation
• Can use these tools on small problems
where we can write out all
combinations
• Next step is to move to larger (but still
finite) sample spaces
• Simplest (but still useful) case is where
all outcomes are equally likely
8. Equally likely events
• S = {e1, e2, e3, ..., em}
• P(ei) = p = 1 / m
• If A ⊂ S, P(A) = #A / m
• Uniformly distributed
9. Examples
• Flipping a coin
• Rolling a dice
• Drawing a card from a shuffled deck
• By definition, whenever we draw
something “at random”
• Sex of an unborn child?
11. Multiplication principle
If the first random experiment (e1) has n1
possible outcomes, and the second
random experiment (e1) has n2 possible
outcomes and results of the first
experiment do not affect the second, how
many possible outcomes does an new
experiment that combines e1 and e2?
13. Sampling with
replacement
• How many possible Texas license
plates are there?
(3 letters followed by three numbers)
(http://www.licenseplateinfo.com/TxChart/pass-tables.html)
• Multiplication principle
15. Sampling without
replacement
• How many possible hands of Texas
holdem? (Two cards drawn from a
pack of 52)
• How many possible community cards
are there? (Five cards drawn from 52)
• We’ll pretend order matters to start
16. Without replacement
• Order matters: permutation
• nPr = n! / (n - r)!
• Order doesn’t matter: combination
• nCr = n! / ( (n - r)! r!)
• Some times simpler to think about the
ones left out.
• Cancel out factorials if doing by hand
