Chapter 5 Probability

1. Section 5.1 Basic Ideas
   • The probability of an event: proportion of successes over attempts
   • Law of large numbers
   • Sample Space: set of all possible outcomes
   • An event
   • Probability model
   • Notation: `P(A)` as the probability of "getting" `A`
   • For any event `A` we have `0 <= P(A) <= 1`
   • Tree diagram -- see fixed example on discretetree.htm,
     or dynamic with replacement on Tree--with Replacement,
     or dynamic without replacement on Tree--without Replacement
   • An unusual event
   • Approximating probabilities by observing data (the Empirical Method)
   • Simulation
2. Section 5.2 The Addition Rule and Rule of Completeness
   • Compound event
   • `P(A or B)` is the probability the `A` or `B` occur (this is the inclusive "or" so it is OK if both occur)
   • `P(A or B) = P(A) +P(B) - P(A and B)`
   • define mutually exclusive
   • If `A` and `B` are mutually exclusive then `P(A and B) = 0`, therefore, in such a case, `P(A or B) = P(A) +P(B)`
   • The complement of an event: complement of `A` is denoted as `A^c` or as `A^'`
   • `P(A^c) = 1 - P(A)` and, of course, `P(A) = 1 - P(A^c)`
3. Section 5.3 Conditional Probability and the Multiplication Rule
   • define conditional probability
   • define unconditional probability
   • Notation: `P( A | B ) ` is the probability of event `A` given that event `B` occurs.
     `P( B | A ) ` is the probability of event `B` given that event `A` occurs.
   • `P(B|A) = (P(A and B))/(P(A))`
   • The General Multiplication Rule
     `P(A and B) = P(A)*P(B|A)`
     but `P(A and B) = P(B and A)` and just following the form that means that
     `P(A and B) = P(B and A) = P(B)*P(A|B)`
   • Independence defined; not being independent is termed dependent
   • By definition, for `A` and `B` independent then `P(A|B) = P(A)` and `P(B|A) = P(B)`
     but that means that for `A` and `B` independent
     `P(A and B) = P(A)*P(B|A) = P(A)*P(B)` and
     `P(B and A) = P(B)*P(A|B) = P(B)*P(A) = P(A)*P(B) = P(A and B)`
   • Sampling with and without replacement, and independence.
   • For samples that are less than 5% of the population, we can ignore the effects of sampling without replacement.
4. Section 5.4 Counting
   • Fundamental Principle of Counting
   • Permutations
   • Definition of factorial: Notation `n! = n(n-1)(n-2)...(3)(2)(1)`
   • Define `0! = 1`
   • Permutations of `n` items taken `r` at a time is
     `quad_nP_r = n(n-1)(n-2) ... (n-r+2)(n-r+1) `

     `quad_nP_r = [n(n-1)(n-2) ... (3)(2)(1)) /((n-r)(n-r-1)(n-r-2) ... (3)(2)(1)`

     `quad_nP_r = ( n! ) / ( (n-r)! )`

   • Combinations
   • Combinations of `n` things taken `r` at a time is

     `((n),(r)) = quad_nC_r = (quad_nP_r)/(r!) = ( (n!)/( (n-r)!))/(r!) = (n!)/( r! * (n-r)! )`

   • `quad_nC_r = quad_nC_(n-r)`