MSDS Spring 2018

DATA 605 Fundamental of Computational Mathematics

Jiadi Li

Week 5 Discussion: Pg.37 Discrete Probability Distributions Ex.19

19.If A, B, and C are any three events, show that P (A \(\cup\) B \(\cup\) C) = P (A) + P (B) + P (C) - P (A \(\cap\) B) - P (B \(\cap\) C) - P (C \(\cap\) A) + P (A \(\cap\) B \(\cap\) C) .

P (A \(\cup\) B \(\cup\) C) = P ( (A \(\cup\) B) \(\cup\) C)
= P (A \(\cup\) B) + P(C) - P( (A \(\cup\) B) \(\cap\) C)
= P (A) + P (B) - P (A \(\cap\) B) + P (C) - P ( (A \(\cap\) C)\(\cup\)(B \(\cap\) C))
= P (A) + P (B) + P (C) - P (A \(\cap\) B) - [ P (A \(\cap\) C) + P (B \(\cap\) C) - P ( (A \(\cap\) B)\(\cap\)(A \(\cap\) B) ) ]
= P (A) + P (B) + P (C) - P (A \(\cap\) B) - P (A \(\cap\) C) - P (B \(\cap\) C) + P (A \(\cap\) C \(\cap\) B)
= P (A) + P (B) + P (C) - P (A \(\cap\) B) - P (B \(\cap\) C) - P (C \(\cap\) A) + P (A \(\cap\) B \(\cap\) C)