probabilty proof question

Asked Oct 31 '19 at 20:10

Active Oct 31 '19 at 20:22

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How do I prove the following:

$\operatorname{Pr}(A \cup B \cup C) = \operatorname{Pr}(A) + \operatorname{Pr}(B) + \operatorname{Pr}(C) $ $\qquad − \operatorname{Pr}(A \cap B) − \operatorname{Pr}(A \cap C) − \operatorname{Pr}(B \cap C) + \operatorname{Pr}(A \cap B \cap C)$?
$\max( \operatorname{Pr}(A), \operatorname{Pr}(B)) \leq \operatorname{Pr} (A \cup B)$

edited Oct 31 '19 at 20:22

Aaron Montgomery

10,688

asked Oct 31 '19 at 20:10

ga as

Welcome to MSE! What have you tried? Where are you stuck? – Aaron Montgomery Oct 31 '19 at 20:15
I understand the logic but im having trouble with the formality of the answer. – ga as Oct 31 '19 at 20:17
1

Can you show us what you do have, so that we can comment on it? Where, specifically, are your concerns with the formality? – Aaron Montgomery Oct 31 '19 at 20:17
This is known as the inclusion/exclusion principle.

Alex R.

Oct 31 '19 at 20:18

regarding questin 2 : i know that pr(A)>=0 and pr(AUB)=pr(A)+pr(B)-A union B – ga as Oct 31 '19 at 20:20

In your last comment, that should be $A \cap B$—that is, "$A$ intersect $B$," not "$A$ union $B$." – Brian Tung Oct 31 '19 at 20:23

right. sorry . How do I formalize the answer? – ga as Oct 31 '19 at 20:26

Regarding (2), you can write it as $$P(A\cup B) = P(A\cup (B\setminus A)) = P(A) + P(B\setminus A)$$ which is the sum of probabilities of disjoint events. – SlipEternal Oct 31 '19 at 20:26

Use Venn diagrams to show set relations. The probabilities will follow trivially. – herb steinberg Oct 31 '19 at 20:50

I cant use Venn diagrams, I have to use the probailty axioms. – ga as Oct 31 '19 at 21:51

could someone please write a full answer for the 2nd question? – ga as Oct 31 '19 at 22:01

probabilty proof question

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