Probability (7) Bayes theorem

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

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

View Full Document Right Arrow Icon
Independence
Image of page 1

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

View Full Document Right Arrow Icon
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.
Image of page 2
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,
Image of page 3

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

View Full Document Right Arrow Icon
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.
Image of page 4
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.
Image of page 5

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

View Full Document Right Arrow Icon
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
Image of page 6
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 )
Image of page 7

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

View Full Document Right Arrow Icon
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
Image of page 8
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?
Image of page 9

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

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern