Probability Rules Chapter 15

Sample Space The sample space of a trial is the set of all possible outcomes and is labeled S. The outcomes do NOT need to be equally likely. Example: To flip a coin twice S={HH},{HT},{TH},{TT}

First Three Rules for Probability Make a Picture

What Kinds of Pictures? Venn Diagram Tree Diagram Contingency Table

Venn Example The probability that a US resident has traveled to Canada is 0.18, to Mexico is 0.09 and to both countries is C ={ Traveled to Canada} M ={ Traveled to Mexico} P(C ) =.18, P(M) =.09 and P(C & M) =.04

Venn Example CM C and M C and M c C c and M C c and M c

Tree Example Twenty percent of cars that are inspected have faulty pollution control devices. 40% of the time a faulty pollution control device costs more that $100 to fix F = faulty device, P(F) =.2 M = more than $100 P(M) =.4

Tree Example F (.2) F c (.8) M (.4) M c (.6) F and M (.2 x.4 =.08) F and M c (.2 x.6 =.12) F c (.8)

Contingency Table Example Pet Store Gender DogsCatsTotal Male Females Totals

Contingency Table Pet Store Gender DogsCatsTotal Male 8/42 6/42 14/ Females 16/42 12/42 28/ Totals 24/42 18/4242/

General Addition Rule P(A or B) = P(A) + P(B) - P(A and B) Works regardless of whether A and B are disjoint If A and B are disjoint then we have P(A or B) = P(A)+ P(B)

Conditional Probability The probability of A given that B has happened is denoted P(A|B). P(A|B) = P(A and B) P(B) P(A|B) is the probability that A happens given that B has already happened

Independent Events Two events A and B are independent, then P(A|B) = P(A) or P(B|A) = P(B) If the probability of A given B is just the probability of A or if the probability of B given A is just the probability of B then A and B are independent

General Multiplication Rule P(A and B) = P(A) x P(B|A) or = P(B) x P(A|B) The probability that A and B happens is the probability that the first happens times the probability that the second happens given that the first has already happened. A and B are not necessarily independent events

General Multiplication Rule If A and B are independent then P(A and B) = P(A) x P(B|A) but P(B|A) = P(B) So P(A and B) = P(A) x P(B) is the events are independent