1. Conditional Probability and Independence Section 3.6

2. Definition 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.

3. 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.

4. Example, Using Table Let E=the sum of the faces is even Let S 2 =the second die is a 2 Find 1. P(S 2 | E) = P(S 2 ∩ E) /P(E)= 3/18=1/6 2. P(E | S 2 )= 3/6=1/2

5. One way of doing this is to construct a table of frequencies: Event EEvent E ’ TOTALS Event S 2 n(E ∩ S 2 )=3n(E’ ∩ S 2 )=3Total S 2 18 Event S 2 ’n(E ∩ S 2 ’)=15n(E’ ∩ S 2 ’)=15Total S 2 ’ 18 Total E = 18Total E ’ =18 Grand Total = 36 Example, Using Table

6. Independence 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.

7. 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 exclusive?