STAT1008 Lecture Notes - Lecture 18: Conditional Probability
STAT1008 Week 6 Lecture C
● Combining Events
○ P(A and B) is the probability that both events A and B will happen <= P(AB)
○ P (A or B) is the probability that either event A or event B will happen (or both)
● Additive Rule: P(A or B)
○ P(A or B) = P(A) + P(B) - P(A and B)
● Complement Rule: P(not A)
○ P(not A) = 1-P(A)
● Conditional Probability
○ P(A if B) is the probability of A, if we know B has happened
○ This is read in multiple ways:
■ Probability of A if B
■ Probability of A given B
■ Probability of A conditional on B
■ You may also see this written as P(A|B) -> Condition on
○ P(A if B) doesn’t = P(B if A)
○ P(A if B) = P(A and B)/P(B)
● Multiplicative Rule: P(A and B)
○ P(A if B) = P(A and B)/P(B)
○ P(A and B) = P(A if B) x P(B)
● Disjoint Events:
○ Events A and B are disjoint or mutually exclusive if only one of the two events
can happen
○ Think of two events that are disjoint, and two events that are not disjoint
○ If A and B are disjoint, then P(A or B) = P(A) + P(B) since P(A and B) = 0 as A
and B are disjoint, then both can’t happen
● Independence:
○ 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
○ Think of 2 events that are independent, and two events that are not independent
○ Pick the best choice:
■ If A and B are independent, then P(A and B) = P(A)P(B)
■ P(A and B) = P(A if B)P(B)
■ If A and B are independent, the P(A if B) = P(A) so P(A and B) = P(A)P(B)
● Disjoint and Independent
○ Assuming that P(A) > 0 and P(B) > 0, then disjoint events are never independent
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Document Summary
P(a and b) is the probability that both events a and b will happen <= p(ab) P (a or b) is the probability that either event a or event b will happen (or both) P(a or b) = p(a) + p(b) - p(a and b) P(a if b) is the probability of a, if we know b has happened. You may also see this written as p(a|b) -> condition on. P(a if b) doesn"t = p(b if a) P(a if b) = p(a and b)/p(b) P(a and b) = p(a if b) x p(b) Events a and b are disjoint or mutually exclusive if only one of the two events can happen. Think of two events that are disjoint, and two events that are not disjoint. If a and b are disjoint, then p(a or b) = p(a) + p(b) since p(a and b) = 0 as a and b are disjoint, then both can"t happen.
