lecture_20

Mon Jul 2 14:00:01 CDT 2012

Ten topic

• Combinatorics including
  • Dice
  • Card Hands
• • Binomial Expansion
  • Geometric Series
  • Exponential Series
  • Logrithmic Series
• • Binomial coefficients
  • Multinomial coefficients
• • Axioms of probability
  • Conditional probability
• Baye's formula
• Independence
• Inclusion-exclusion
• • Expectation
  • Variance
• • Binomial random variable [(n,p)]
  • Poisson random variable [(\lambda)]
  • Geometric random variable [(p)]
• • Stirling's Formula
  • Integrals of probability theory

Practice Problems

Let [X] be the r.v. "# of head in 3 tosses (fair coin)"

• [P( X = 3 | X \ge 1)]
  • [\frac{ P( X = 3)}{ P( X \ge 1)} = \frac{ \left(\frac{ 1}{ 2}\right)^3}{ 1 - \left(\frac{ 1}{ 2}\right)^3 = \frac{ \frac{ 1}{ 8}}{ \frac{ 7}{ 8}} = \frac{ 1}{ 7}]
• [P(X = 3)]

Practice Problem

Drawing a card at random from a certain deck of cards. It might be a standard deck. But there is a [\frac{ 1}{ 3}] probability that one "5" is missing. What is the probability you draw a "5"?

[ \frac{ 2}{ 3} \frac{ {4 \choose 1}}{ {52 \choose 1}}

• \frac{ 1}{ 3} \frac{ {3 \choose 1}}{ {51 \choose 1}}]