The Addition Rule and Complements

5.2

●Venn Diagrams provide a useful way to visualize probabilitiesThe entire rectangle represents the sample space S

The circle represents an event E

VENN DIAGRAMS

S

E

In the Venn diagram belowThe sample space is {0, 1, 2, 3, …, 9}The event E is {0, 1, 2}The event F is {8, 9}The outcomes {3}, {4}, {5}, {6}, {7} are in neither event E nor event F

VENN DIAGRAM

●Two events are disjoint if they do not have any outcomes in common●Another name for this is mutually exclusive●Two events are disjoint if it is impossible for

both to happen at the same time●E and F below are disjoint

MUTUALLY EXCLUSIVE

●For disjoint events, the outcomes of (E or F) can be listed as the outcomes of E followed by the outcomes of F●There are no duplicates in this list●The Addition Rule for disjoint events is

P(E or F) = P(E) + P(F)

●Thus we can find P(E or F) if we know both P(E) and P(F)

ADDITION RULE

●This is also true for more than two disjoint events● If E, F, G, … are all disjoint (none of them

have any outcomes in common), thenP(E or F or G or …) = P(E) + P(F) + P(G) + …●The Venn diagram below is an example of this

ADDITION RULE

●In rolling a fair die, what is the chance of rolling a {2 or lower} or a {6}The probability of {2 or lower} is 2/6The probability of {6} is 1/6The two events {1, 2} and {6} are disjoint

●The total probability is 2/6 + 1/6 = 3/6 = 1/2

EXAMPLE

The addition rule only applies to events that are disjoint

If the two events are not disjoint, then this rule must be modified

WHAT IF NOT DISJOINT?

The Venn diagram below illustrates how the outcomes {1} and {3} are counted both in event E and event F

VENN DIAGRAM

●In rolling a fair die, what is the chance of rolling a {2 or lower} or an even number?The probability of {2 or lower} is 2/6The probability of {2, 4, 6} is 3/6The two events {1, 2} and {2, 4, 6} are not disjoint

The total probability is not 2/6 + 3/6 = 5/6

The total probability is 4/6 because the event is {1, 2, 4, 6}

EXAMPLE

●For the formula P(E) + P(F), all the outcomes that are in both events are counted twice●Thus, to compute P(E or F), these outcomes

must be subtracted (once)●The General Addition Rule is

P(E or F) = P(E) + P(F) – P(E and F)

●This rule is true both for disjoint events and for not disjoint events

GENERAL ADDITION RULE

●When choosing a card at random out of a deck of 52 cards, what is the probability of choosing a queen or a heart?E = “choosing a queen”F = “choosing a heart”

●E and F are not disjoint (it is possible to choose the queen of hearts), so we must use the General Addition Rule

EXAMPLE

5216

521

5213

524

heart)and(queen

(heart)(queen)heart)or(queen

P

PPP

EXAMPLE

P(E) = P(queen) = 4/52P(F) = P(heart) = 13/52P(E and F) = P(queen of hearts) = 1/52, so

The Probability of an event with the word AND must have that event in ALL of the experiments.

Example:A = 1,3,5,7

B = 2,3,5So, The outcomes of A and B are 3 and 5

THE WORD AND

ExampleIf A = 1,2,3

B = 4, 5Find P (A and B)Empty Set written { } or Ø

EMPTY SET

●The complement of the event E, written Ec, consists of all the outcomes that are not in that event●Examples

Flipping a coin … E = “heads” … Ec = “tails”Rolling a die … E = {even numbers} … Ec = {odd numbers}

Weather … E = “will rain” … Ec = “won’t rain”

COMPLEMENT

●The probability of the complement Ec is 1 minus the probability of E●This can be shown in one of two ways

It’s obvious … if there is a 30% chance of rain, then there is a 70% chance of no rain

E and Ec are two disjoint events that add up to the entire sample space

PROBABILITY OF A COMPLEMENT

The Complement Rule can also be illustrated using a Venn diagram

VENN DIAGRAM

Probabilities obey additional rulesFor disjoint events, the Addition Rule is

used for calculating “or” probabilitiesFor events that are not disjoint, the

Addition Rule is not valid … instead the General Addition Rule is used for calculating “or” probabilities

The Complement Rule is used for calculating “not” probabilities

SUMMARY