ENGLISH
Probability (7) Bayes theorem

# Probability (7) Bayes theorem - Independence Irrelevant...

• Notes
• 41

This preview shows pages 1–10. Sign up to view the full content.

Independence

This preview has intentionally blurred sections. Sign up to view the full version.

Irrelevant information Based on a new information, we can update the sample space and events to be new ones, so that we can calculate a conditional probability of the updated event. However, sometimes, the new information may be irrelevant. Therefore, the conditional probability of B given A, P(B|A), is exactly the same as the unconditional probability of B. That is, we do not obtain any “new” relevant information from the event A to update the probability of B. In such a case, we would say that B and A are (statistically) independent . If two events are NOT independent, then we would say that they are dependent.
Independence ) ( ) ( ) ( B P A P B A P = Definition (Independence) ) ( ) | ( B P A B P = 0 ) ( A P when Two events A and B are called (statistically) independent if and ony if OR, equivalently,

This preview has intentionally blurred sections. Sign up to view the full version.

Theorem: If events A and B are independent, then all of A and B C , A C and B , and A C and B C are also independent. Independence Prove it later.
Example On page 25 of the course note Question 14: A fair coin is tossed three times. Denote the event that a head occurs on each of the first two tosses by A , the event that a tail occurs on the third toss by B , and the event that exactly two tails occur in the three tosses by C , show that 1) Events A and B are independent; 2) Events B and C are dependent.

This preview has intentionally blurred sections. Sign up to view the full version.

Example 8 3 ) ( , 2 1 ) ( , 4 1 ) ( = = = C P B P A P On page 25 of the course note Denote the event that a head occurs on each of the first two tosses by A , the event that a tail occurs on the third toss by B , and the event that exactly two tails occur in the three tosses by C , Events A and B are independent? ? ) ( = B A P
Example 8 3 ) ( , 2 1 ) ( , 4 1 ) ( = = = C P B P A P On page 25 of the course note Denote the event that a head occurs on each of the first two tosses by A , the event that a tail occurs on the third toss by B , and the event that exactly two tails occur in the three tosses by C , Events A and B are independent? 8 1 ) ( = B A P =P( A ) X P( B )

This preview has intentionally blurred sections. Sign up to view the full version.

Example 8 3 ) ( , 2 1 ) ( , 4 1 ) ( = = = C P B P A P On page 25 of the course note Denote the event that a head occurs on each of the first two tosses by A , the event that a tail occurs on the third toss by B , and the event that exactly two tails occur in the three tosses by C , Events B and C are independent? ? ) ( = B C P
Example 8 3 ) ( , 2 1 ) ( , 4 1 ) ( = = = C P B P A P On page 25 of the course note Denote the event that a head occurs on each of the first two tosses by A , the event that a tail occurs on the third toss by B , and the event that exactly two tails occur in the three tosses by C , Events B and C are independent?

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern