Download presentation

Presentation is loading. Please wait.

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?

Similar presentations

© 2024 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google