11Bcomplementary


 * Complementary Events **


 * Two events are ** complementary ** if either one event will happen, or the other.
 * both events can't happen at once
 * it is impossible for neither event to happen
 * Recall the Venn Diagram that shows the Complement of A


 * If A is one event, then A' is the complement of that event


 * Pr(A) + PR(A') = 1
 * this can be rearranged to give: Pr(A') = 1 – Pr(A)


 * Pr(A Ç A') = 0
 * probability of both A __**and**__ A' happening is 0


 * Pr(A È A') = 1
 * probability of either A __**or**__ A' happening is 1


 * ** Example 1 **
 * a 6 sided dice is rolled
 * Let A be the event that a 1 or 2 occurs
 * Let B be the event that a 2, 4, 5 or 6 occurs

math . \qquad \qquad \qquad Pr(A \, \cap \, B) = \dfrac{0}{6} = 0 math . math . \qquad \qquad \qquad Pr(A \, \cup \, B) = \dfrac{2}{6} + \dfrac{4}{6} = \dfrac{6}{6} = 1 math


 * Because of these two results, A and B must be complementary


 * ** Example 2 **
 * Rain tomorrow and no rain tomorrow are complementary events
 * so
 * If Pr(rain tomorrow) = 0.2
 * Then Pr(no rain tomorrow) = 1 – 0.2 = 0.8


 * Mutually Exclusive Events **


 * Two events are ** mutually exclusive ** if the two events can't happen at once
 * Mutually exclusive events don't have to be complementary
 * Pr(A È B) __<__ 1
 * All Complementary events are mutually exclusive


 * ** Example 3 **
 * A 6 sided dice is rolled
 * Let A be the event that a 4 occurs
 * Let B be the event that a 6 occurs


 * A and B are mutually exclusive because they can't both occur at once
 * Pr(A Ç B) = 0

math . \qquad \qquad \qquad Pr(A \, \cup \, B) = \dfrac{2}{6} = \dfrac{1}{3} math
 * A and B are not complementary because Pr(A È B) is not equal to 1.

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