Important to distinguish independence from mutually exclusive which would say B ? A is empty (cannot happen). Example. Deal 2 cards from deck. A first card is
If A and B are mutually exclusive then P(A ? B) = P(A) + P(B). This means events A and B cannot happen together. If A happens
experiment then the conditional probability of the event E under the Three events A
(b) Draw a Venn diagram summarizing the variables and their associated 2) Using Bayes' Theorem: If the two events are independent then P(below PL.
If A and B are mutually exclusive then P(A ? B) = P(A) + P(B). This means events A and B cannot happen together. If A happens
https://faculty.math.illinois.edu/~hildebr/370/370generalproblemssol.pdf
1. If two events (both with probability greater than 0) are mutually exclusive then: A. They also must be independent. B
other values of k (for example if k>n the probability is 0 since then (b) The event A > B is independent of the event B > C since A > B is the same.
If A and B are independent events with P(A)=0.6 and P(B)=0.3 find the following: (a) P(A U B) (b) P(A n B) (c) P(A U B ) (d) P(A
20 janv. 1992 103/3.2 As provided in Regulation 3 b) officials shall not be subject ... of the salary of the official or
Theorem 2 (Conditional Probability of Independent Events) If A and B are independent events with nonzero probabilities in a sample space S then P(A jB) = P(A); P(B jA) = P(B): If either equation in (4) holds then A and B are independent Example 3 A single card is drawn from a standard 52-card deck Test the following events for independence:
De nition 1: Independent Events The occurrence of one event has no e ect on the probability of the occurrence of any other event Events A and B are independent if one of the following is true: (1) P (AjB) = P (A); (2) P (BjA) = P (B); (3) P (AandB) = P (A)P (B) De nition 2: Mutually Exclusive
any two events A and B with P(B) > 0 the conditional probability of A given that B has occurred is de?ned by P(AB) = P(A? B) P(B) Furthermore two events are independent if any one of the following is true: P(A?B) = P(A)P(B) P(AB) = P(A) P(BA) = P(B) Exercise2 7(Conditional Probability onthe Independence of Events) 1
P[A? B]= P[A? C]= P[B? C]= 1 36 so that all events are pairwise independent However P[A? B? C]= P[B? C]= 1 36 while P[A]P[B]P[C]= 1 216 so they are not independent as a triplet First note that indeed P[A? B] = P[B? C] = 1 36 since the fact that A and B occurred is the same as the fact that B and C occurred Example 2
Events A and B are independent if: knowing whether A occured does not change the probability of B Mathematically can say in two equivalent ways: P(BA)=P(B) P(A and B)=P(B ? A)=P(B) × P(A) Important to distinguish independence from mutually exclusive which would say B ? A is empty (cannot happen) Example
1 If two events (both with probability greater than 0) are mutually exclusive then: A They also must be independent B They also could be independent C They cannot be independent 2 If two events (both with probability greater than 0) are mutually exclusive then: A They also must be complements B They also could be complements C