Stats 1430 Chapter 5 Probability Rules and Conditional Probability
Seeing AD and making purchase
Q: 40% of customers saw the Ad, and of those, 30% made a purchase. Of those who didn't see the ad, only 20% made a purchase A: P(Ad)=40% P(P|Ad)=30% P(P|NoAd)=20% Q: What % of all customers saw the Ad and made a purchase? 1)Use M. Rule 2)Fill out the other "AND" probabilities on a two way table A: P(Ad and P)=P(Ad)*P(P|Ad)=0.4*0.3=.12
"AND" Probabilities: Multiplication Rule
-For two events A and B, the probability of both events occurring is: P(B|A) = P(B and A)/P(A) -OR... P(A and B)= P(A)*P(B|A) -Represents the results for the group A, weighted by % in group A -Example: Y N Tot F .15 .45 .60 M .12 .28 .40 Tot .27 .73 1.00 P(Y|F) = P(Y and F)/P(F) = .15/.60 = .25 OR P(Y and F) = P(F)*P(Y|F) = .60*.25 = .15 *60% are female & 25% of those said yes
Do you live on campus? (Independent or Dependent)
-Select 2 students at random and ask: Do you live of campus? Q:Their outcomes are independent Why? A: Random Sample = Independent
Gender and living off campus (Independent or Dependent)
-Suppose 60% of male students live off campus and 60% of female students live off campus Q:Gender and living off campus are independent Why? A: The percentages equal each other so no relationship. Gender does not affect living off campus. P(Off|M) = P(Off|F) = P(Off) = 0.60
Basic Terms of Probability
Sample Space ~A list of all possible outcomes ~S={all possible outcomes} ~S={1,2,3,4,5,6} Events ~Characteristics ~Subset of S; A,B,C, etc ~A={Event} Probability of an event ~Characteristics ~P(A)="Pot A"=long term chance of event A happening
3 Results that are true is and only if A and B are independent
1) P(A|B) = P(A) (Can switch A and B) a. Actual definition of independence b. Conditional = Marginal 2) P(A|B) = P(A|Not B) a. Two conditionals are equal 3) P(A and B) = P(A)*P(B) a. If you know its independent then this is true
Types of Probability
1)Marginal probability, P(A) ~Probability of a single event (characteristic) ~Ex: The % of Democrats = P(D) 2)Conditional Probability ~P(B|A) = Probability of B, given A has occurred ~Means the % of group A that have characteristic B ~The % of democrats who are male =P(M|D) ~The % of male democrats =P(M|D) 3)"And" probability ~P(A and B) = Probability both A and B ~AKA "Joint Probability" ~Intersection of A and B ~Concave down parabola 4)"Or" probability ~P(A or B) = Probability A or B or Both occur ~Probability that at least on of them occurs ~Union of A and B ~Concave up parabola
Special Case: Independent Events
Def: Two events (or characteristics) A and B are said to be independent if their outcomes don't effect each other P(A&B) = P(A)*P(B|A) Whether A occurred doesn't affect P(B) ~P(B) = P(B|A) Whether B occurred doesn't affect P(A) ~P(A) = P(A|B) So if independent, simplify M. Rule ~P(A&B) = P(A)*P(B|A) = P(A)*P(B)
Two-Way Table (Full tank of gas using cash or credit)
Full Tank Partial Tank Total Cash P(F&C)=0.1 P(P&C)=0.2 P(C)=0.3 Cr-C P(F&CC)=.25 P(P&CC)=.45 P(CC)=.7 Q: Probability of someone who gets a full tank pays cash? A: P(Full|Cash) = 0.1/0.3 Q: Probability of someone who pays cash gets a full tank? A: P(Cash|Full) = 0.1/0.35 P(What you want to know | already know)
