STAT1008 Study Guide - Final Guide: Bayes Estimator, Venn Diagram, Conditional Probability
Probability
11.1 Probability Rules
• Event – soethig that either happes or does’t happe is true or is ot true.
• Probability of event (A) is the long-run frequency or proportion of times the event occurs.
• Probability always between O and 1.
• P(A) = 1 means A will definitively happen.
• P(A) = 0 means A will definitively not happen.
• If there are equally likely outcomes, then P(A) = no. of outcomes of event A/total no. of
outcomes.
Combination of Events
• P (A and B), or P (A or B).
Additive Rule
• P (A or B) = P(A) + P(B) – P(A&B).
• Use Venn diagram.
Complement Rule
• P (not A) = 1 – P(A).
• P (not (A or B)) = 1 – [P(A) + P(B) – P(A and B)]
Conditional Probability
• P (A if B) = P(A and B)/P(B), is the probability of A, if we know B
has happened.
• You may also see this written as P (A I B).
• This is read i ultiple ays: proaility of A if B, proaility
of A gie B or proaility of A oditioal o B.
• Note: P (A if B) ≠ PB if A.
Multiplicative Rule
• P (A and B) = P (A if B) P (B).
• Equivalent form: P (A and B) = P (A) P (B if A).
Special Case: Disjoint
Events
• Events A and B are disjoint or mutually exclusive if only one of
the two events can happen.
• If A and B are disjoint, then P (A or B) = P (A) + P(B).
• If A and B are disjoint, then both cannot happen, so P(A and B) =
0.
Special Case:
Independent Events
• Events A and B are independent if P (A if B) = P (A).
• Intuitively, knowing that event B happened does not change the
probability that event A happened.
• If A and B are independent, then P (A and B) = P(A)P(B).
11.2 Tree Diagrams and Bayes' Rule
Total Probability Rule
• For any two events A and B, P(B) = P(A and B) + P(not A and B) = P(A)P(B if A) + P(not A)P(B if
not A).
• If events B1, B2, through Bn are disjoint and together make up all possibilities, then: P(A) = P(A
and B1) + P(A and B2 + … + PA ad Bn).
Tree Diagrams
• The initial set of branches show the probabilities from one set of events and the second set of
branches show conditional probabilities.
• Multiplying along any set of branches uses the multiplicative rule to find the joint probability
for that pair of events.
Bayes' Rule
If A and B are any two events:
• P(A if B) = P(A and B)/P(B)
• Bayes’ Rule arious fors:
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
If there are equally likely outcomes, then p(a) = no. of outcomes of event a/total no. of outcomes. If a and b are disjoint, then p (a or b) = p (a) + p(b). If a and b are disjoint, then both cannot happen, so p(a and b) : events a and b are independent if p (a if b) = p (a). Intuitively, knowing that event b happened does not change the probability that event a happened. If a and b are independent, then p (a and b) = p(a)p(b). For any two events a and b, p(b) = p(a and b) + p(not a and b) = p(a)p(b if a) + p(not a)p(b if not a). If events b1, b2, through bn are disjoint and together make up all possibilities, then: p(a) = p(a and b1) + p(a and b2(cid:895) + + p(cid:894)a a(cid:374)d bn).
