Conditional Probability and IndependenceDefinitionExercise #1Example, Using TableSlide 5Independence of eventsIndependent EventsConditional Probability and IndependenceSection 3.6Definition•A conditional probability is a probability whose sample space has been limited to only those outcomes that fulfill a certain condition. •The conditional probability of event A given that event B has happened is P(A|B)=P(A ∩ B)/P(B).•The order is very important do not think that P(A|B)=P(B|A)! THEY ARE DIFFERENT.Exercise #1•Suppose that A and B are events with probabilities: P(A)=1/3, P(B)=1/4, P(A ∩ B)=1/10•Find each of the following: 1. P(A | B) = P(A ∩ B)/P(B)=1/10/1/4=4/10 2. P(B | A) = P(A ∩ B)/P(A)=1/10/1/3=3/10 3. P(A’ | B’) = P(A’ ∩ B’)/P(B’)= P((A U B)’)/(1-P(B))=(1-P(A U B))/(1 – P(B))= (1 – (P(A)+P(B)-P(A ∩ B)))/(1-P(B))= (1 – (1/3+1/4-1/10))/(1-1/10)=(1-29/60)/9/10= 31/60/9/10=31/54.Example, Using Table•Let E=the sum of the faces is even•Let S2=the second die is a 2•Find 1. P(S2 | E) = P(S2 ∩ E) /P(E)= 3/18=1/6 2. P(E | S2)= 3/6=1/2 1,1 2,1 3,1 4,1 5, 1 6,11, 2 2, 2 3, 2 4, 2 5, 2 6, 21,3 2,3 3,3 4, 3 5, 3 6, 31, 4 2, 4 3, 4 4, 4 5, 4 6, 41, 5 2,5 3,5 4,5 5,5 6,51, 6 2, 6 3, 6 4, 6 5, 6 6, 6S •One way of doing this is to construct a table of frequencies:Event EEvent E’TOTALSEvent S2n(E ∩ S2)=3 n(E’ ∩ S2)=3 Total S2 18Event S2’ n(E ∩ S2’)=15 n(E’ ∩ S2’)=15 Total S2’18Total E = 18Total E’ =18Grand Total = 36Example, Using TableIndependence of events•Two events E and F are said to be independent if and only if P(E ∩ F)=P(E)P(F).•If the above condition is not satisfied, then we say the two events E and F are dependent.•When we say two events are independent, we are saying that if event E has occurred, this will not effect the probability of event F.•INDEPENDENT EVENTS: The occurrence of one event has no effect on the probability of the other.Independent Events•Consider flipping a coin recording the outcome each time. Are these events independent????•You throw 2 fair dice, one is green, one is red. Observe the outcomes.•Let A be the event that the sum is 7 •Let B be the event that the red die shows an even number •Are A and B Independent?•Are A and B Mutually
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