Week 5

• sample space (S)


  the set of all possible outcomes of a random phenomenon
• event

  subset of the sample space S
• union ($A \cup B$)


  the event that consists of all outcomes that belong to A or B or both

  
• intersection ($A \cap B$)


  the event that consists of all outcomes common to A and B

  
• complement of A ($A^c$)


  consists of all outcomes that are not in A

  $A +$$A^c =$$1$
• mutually exclusive/disjoint events


  events that can never occur simultaneously

  $A \cap B =$$0$
• probability of an event

  the proportion of times that the event occurs in a long run of trials

  • always between 0 and 1
• axiom 1

  $P(A) \ge 0$
• axiom 2


  $P(S) =$$1$
• axiom 3

  if A and B are mutually exclusive,

  $P(A \cup B) =$$P(A) +$$P(B)$
• axiom 4

  if $A_1, A_2, ... , A_r$are pairwise mutually exclusive (no two can occur simultaneously), then

  $P(A_1 \cup A_2 ... \cup A_r) =$$P(A_1) +$$P(A_2)+$$... +$$P(A_r)$
• additive law of probability (axiom 5)

  for any 3 events A, B, C:

  $P(A\cup B \cup C) =$$P(A) +$$P(B) +$$P(C) - P(A\cap B) - P(A\cap C) - P(B \cap C) +$$P(A \cap B \cap C)$
• implications of the axioms

  $P(A^c) =$$1 - P(A)$

  $P(A \cup B) =$$P(A) +$$P(B) - P(A \cap B)$

  $P(A) =$$P(A \cap B) +$$P(A \cap B^c)$
• when all outcomes in an experiment are equally likely

  $P(A) =$$\frac{\text{number of outcomes in A}}{\text{total number of possible outcomes in S}}$
• two events A and B are called independent events if


  $P(A \cap B) =$$P(A)P(B)$
• can mutually exclusive events be independent

  no because we know that if A happens, B for sure doesn't happen with mutually exclusive events
• conditional probability of A given B


  $P(A |B) =$$\frac{P(A \cap B)}{P(B)}$ when $P(B) > 0$
• partitioning a sample space

  • suppose that $B_1, B_2,... , B_n$are mutually exclusive events within a sample space, and that $B_1 \cup B_2 \cup ... \cup B_n =$$S$
  • then $B_1, B_2,... , B_n$are said to partition the sample space S
  • for any event A, we have $P(A) =$$\sum_{i = 1}^nP(A \cap B_i)$
• Bayes theorem

  two events, A and B, on the same sample space S where P(A) > 0

  $P(B |A) =$$\frac{P(B)P(A|B)}{P(A)}$
• prevalence of a disease


  number of people who currently have the disease divided by the number of people in the population
• sensitivity


  the probability that the test is positive, given that the person has the disease

  $\text{sen} =$$P(+|D)$
• false positive


  test shows positive result for a non-disease subject

  $P(\text{false positive}) =$$P(+|D^c)$
• specificity


  the probability that the test is negative, given that the person does not have the disease

  $\text{spec} =$$P(-|D^c)$
• false negative


  test shows negative result for a diseased subject

  $P(\text{false negative}) =$$P(-|D)$